I've watched both animes fully, namely Naruto twice and One Piece twice as well, so I believe I am somewhat qualified to touch on this topic. I have read some of the discussions and watched videos on this topic but none of them seems satisfying in the depth of discussion. So I decided to write my own opinion and I will try to be as objective as I can. It's my first time writing, so hope you guys could at least be gentle with critiques. By the way, it going to be long.
First of all, I am going to layout my structure of analysis. I know there both anime have huge number of fans, so they are both successful and which one is better will depend on taste. But this doesn't mean that there is no room for comparison. This analysis will start with the topic each author has chosen and their consequences, followed by major themes, plot design and character development, then at last end with some personal preferences and comments. Please be aware that the content displayed will disclose some important plots, so if you don't what to be spoiled, don't read it. So lets begin!
1.TOPIC CHOICE AND ITS CONSEQUENCE
Naruto's author Masashi Kishimoto (MK) picked ninjas as the main topic and used a lot of Japanese legends or folk tales in his story. Ninjas were traditionally careers in darkness, responsible for assassination and spying, but MK brought the ninjas under light and described as the most wonderful career one could ever have. To people outside of Japan, ninjas are mysterious and cool, a very unique part of Japanese culture. Since it is little known by outsiders, mainly westerners, the topic itself generates higher interest. Some may also be influenced by The Ninja Turtles who are more like being like western superheros with a title named ninja and thought that ninjas were heros, which historically they were not. So just for the topic itself, MK's ninja topic, in general gains more interest and curiosity on foreign lands. This interest is further increased by the creation of chakra and allows all the moves in the story to be more fancy. At least, when watching the first few episodes, where characters are able to duplicate themselves is more attractive then just throwing punches.
One Piece's author Eiichiro Oda (Oda), on the other hand picked a more familiar topic to foreign people, pirates, especially westerners. Therefore, the anime would not have caught that much attention compared to Naruto. The story starts will our main character being somewhat interesting because he is a rubber man and can stretch. Then we know that in the OP world, there is something called a devil fruit which can let people obtain powers. Its fights are mostly physical, therefore somewhat plain in the beginning. With these basic actions set, it also makes Naruto's video games on average more popular than One Piece, because of more fancy big moves to play. There is of course the most obvious reason, graphics. Naruto's graphics, in general, have a better sense of beauty, which makes it more attractive to get in and start watching. One Piece, because of the story in whole cannot draw its characters with that sense of beauty, or else it would seem strange. So with the discussion above, Naruto achieved better on getting people outside of Japan to start watching, which in turn explains why its popularity on foreign lands are higher than One Piece.
Please note I do not mean that Naruto is better. I am just trying to explain why Naruto is more popular on foreign lands.
2. THEME & PLOT DESIGN
Naruto's main plot is about the main character, Naruto, dreaming of becoming a Hokage, which is the respected leader of his village, and strongest ninja of all. The story starts with a very attractive setting: Naruto has an ambitious dream but is regarded as danger because of Nine-tail fox sealed inside of him and he is also considered to have no talent among children his age. With isolation and having no friends, the situation of Naruto easily creates sympathy among readers and viewers because this topic is so common. Everyone fears isolation and wants to have friends, especially for kids and teens, the majority of anime viewers. Also, being judged by others as lame, and being laughed at by fellow classmates are almost universal worries for people in school. This was portrayed very early in the story and gain better attraction. So we are moved by Naruto when he gets recognition and makes friends; we admire his courage and will to never give up.
After watching the first few dozen episodes or so, Naruto seems deep. It starts to portrait power conflicts inside the village and between countries. The world structure for Naruto is not very grand, just surrounding five countries. The smaller world setting allows more depth of conveying the themes of misunderstanding, hatred, and destiny. On the other side, we have Naruto, the warm hearted protagonist with themes of friendship, hard work and kindness. A very essential part in Naruto is conveying the messages of inherited will and different paths chosen to pursue the same dreams. One the two typical representation of these messages are Madara and Hashirama who both want eternal peace but the former wants to create a flawless world by hypnotizing the whole population, while the latter tries to build understanding among nations. Inherited will can be seen from Naruto and Ashura, Sasuke and Itachi.
However, the one major feature that differentiates Naruto from other mainstream anime is that almost every character initially perceived as evil is "whitewashed", explaining why the evil character developed to this state, his or her sufferings, which makes it hard to be fully angry at them. MK might be trying to use this feature to deliver the message that humans are inherently good, but may be changed by external events and become evil. But, this decreases the satisfaction of watching the good side win and sometimes even creates a sorry feeling for the evil side that lost. In order to "wash" characters, Naruto also involves a lot of long flashbacks, which in a way disrupts the flow of the whole story. Some flashbacks do have really close and meaningful connections to the current act. This was mostly seen prior to the World war.
The major tool for "whitewashing" is preach. There are many preaches the series and Naruto does the most. Changing Gaara to understand love, persuading Nagato or Pain to believe in himself, and converting Tobi to his side, Naruto had some very epic speeches in the series. However, sometimes the logic is not very coherent, and it seems that MK over exaggerated Naruto personal charm. Characters who originally had a very firm worldview can be converted to the Naruto's view of the world by a fight and Naruto's persuasion. The only character that fails is Sasuke. Sasuke is somewhat intertwined with both the plot design and character development, so I would save that comment for later.
One of the biggest changes in Naruto was its fighting settings. In the first stage, battles were more about teamwork and strategy and that is why the characters in school are organized into groups of three. However, as the story progresses and more of nine tailed beast were involved, fighting leaned towards power based. From Pain to World war, the powers of Naruto, Sasuke, Madara, and a handful of other individuals became so big that all the strategic battling stated before was disregarded.
Overall, the anime tries to allude to the real world, like misunderstanding between nations, conflicts of power. These themes gave the story a greater depth, but in later parts of the series, "whitewashing" was not very successful because some of the reasons stated by the characters of why they turn good were not very satisfying. Some of the characters over reacted to certain event and did not make much sense of why one would do that. The first stage was, in general, more coherent and logical than the second stage, Shippuden, as it focuses mainly school tests and conflicts within and nearby the village. The series in a whole is continuous because there is no real end of an event until the end of the story; events are intertwined and the main antagonist is never too clear in the story. It also uses tragic events to create suspense throughout the series.
-- One Piece
At a glance, One piece may be more childish than Naruto because the series contain a lot of comedy. However, after understanding a few arcs of the story, it is much more deeper than what it may seem. The more obvious themes of One Piece are people's dreams, the flow of time and the inherited will. This has been seen in many of the arcs, most notable in the Skypiea arc. But One Piece is far more deeper than that. It involves authoritarian of the World Gov't, balance of power, discrimination among different races, corruption of authorities, different interpretations of freedom and more. The world combined all the forms of production since the start of human history, slavery, enfeoffment, and capitalism. However, these hidden themes are covered by comedy and Straw hats adventures. Oda rarely uses speeches from main characters to convey all of these themes; instead he uses a whole story to tell the messages.
For the plot design, One piece is much more organized than Naruto. Each arc is relatively independent from each other and are attractive stories on their own, but each arc provides some important build ups for the next. Emotions accumulate. For example, the funeral of Merry is considered one of the most moving moments of the series, but with just watching the Water Seven Arc, it is not possible to feel that intense emotion. Every arc also provides some hints on what the will of D is and the 100 years of wiped out history. Every flashback is short and gives relevant background information. There are many hidden mysteries in the series and lots of foreshadowing. Unlike Naruto where becoming a Hokage is more a like happily ever after ending note, One Piece has placed many implications for being Pirate King and which kind. Luffy and Blackbeard of different interpretations and what do they imply? These are suspense in One Piece. Adventures are straight forward since we know the boss very early in each arc but each story pushes all the other messages a little closer to their answers. Moreover, in One Piece, characters never die to just create tragic moments, they die for a reason. Whitebeard's death represents the fall of the old era and the beginning of a new era. Ace's death is for Luffy to truly understand that he is weak and needs to become stronger. A maturer Luffy would be essential to the story afterwards.
For the settings, One Piece is also much more consistent than Naruto. Devil fruits powers are limited to one person each (for the exception of Blackbeard, but that is one of the mysteries in the story, none of the basic settings change). Haki, on the other hand, appeared rather early in the series but was formally introduced during Rayleigh's lecture. Oda, already had a concept of how to deal with logia type devil fruit. Unlike Naruto, One Piece never allowed Luffy's power to grow too fast. After learning sage mode, Naruto was almost the strongest in the village, then after controlling nine-tails, his power was so great that other side characters seemed like nothing. On the other hand, after two years of training, Luffy was much stronger than before, but he still seems to be slightly weaker than admirals, which is more plausible considering that he was just learning the fundamentals. So though the powers of the character later in the story are stronger, they don't make previous character seem like nothing.
Overall, for plot design and themes, Oda is in much greater control than MK. It has been said that Oda has the main plot and the ending of One Piece when he started drawing, while MK seems not so clear about how Naruto will end, so had to use a lot of flashbacks and side events in trying to keep the story coherent.
I will finish the second half tomorrow or later
I've read and watched both series and I love both of them.
That being said, One Piece, in my opinion, is just overall better. From world building, to plot, to characters, to just everything. I love One Piece so i'm a bit biased. But its just my opinion, One Piece is life xD