"It's about to be a girl fight!"
First of all I want to make it clear that I think both are amazing characters and I love them to death. I'm going to keep an open mind about both of them when comparing them. They both have things that they did better than the other. So who's the better character? We'll just have to see.
What they have in common
I think it's fair to first mention what they have in common. They're both redheads, adventurous, curious, wild, whiny, selfish, have parent issues, rebellious, brave, heroic, free-spirited, make deals with witches,and of course cause a big mess with all their mistakes. Let's also not forget that they're usually considered bad role models and are criticized a lot.
What's different about them
Well Ariel's thing is that she falls for a good looking guy and tries her hardest to be with him no matter what. However Merida is actually trying her hardest to avoid getting married. That's definitely a HUGE difference between characters. Now it's time to get to the really fun part of comparing the characters, telling who did what better.
Reasons for rebelling
Merida's rebellion is mostly to avoid getting married but it's shown that she's been rebellious even before that. She doesn't like being a princess and hates all those duties and lessons she has. Ariel on the other hand, though not as obvious, doesn't care much for being a princess either. However her main reason for rebellion is for the injustice her father and all sea creatures have against humans. What gets me about this character is that she fights against injustice just like Pocahontas and Esmeralda but unlike them she's actually rebelling to defend not only a different culture but a completely different species that she knows nothing about. Sure she's aware some are bad but she sees them like with fish, some are good and some are bad. Merida's reasons for rebelling against having to get married are good and strong too, especially when she takes action during the archery contest, but for me Ariel's are stronger.
Being whiny and explaining
Ariel's reasons for being whiny is that she wants to become human because she doesn't fit in and is fascinated by the human world. She even has a song where she's whining about what she wants. Merida whines about how she just doesn't want to be a princess or get married. They both also fight with their parents while being a bit whiny, well at least Ariel is. Merida is way too over the top by saying her mom was never there for her, that this is so unfair, and she'd rather die than be like her mother. Yeah.... that's a... way over the top. Plus it seems with Merida all she does is just whine but doesn't try to explain to her mother why it's unfair and why she's not ready to get married, she just whines and acts all bratty when she could just explain. Ariel was also whiny and at times is her own little drama queen. However she does at least try to explain to her father, he just won't let her explain. So for this I'd say Ariel wins this round.
Both princesses are quite brave and heroic first of all. Ever since Brave was first being shown in commercials, trailer, and posters they've been advertising Merida as a total kicka** character who saves the day and can take care of herself, that's sadly not the case. She has come bada** moments, like when she does archery and when she saves her mother but that's pretty much it. For the rest of the time she usually has to be saved and is in distress where she can't handle herself. Such as when she first meets Mordu and her mom needs to save her, when Mordu comes after her and her mom saves her again, she gets locked in a room and her little brothers have to save her, and during the climax her mother has to save her again. I wouldn't exactly call her weak or a complete damsel in distress but from what they advertised I thought she'd be the next Mulan and Kim Possible.
Ariel on the other hand always gets herself into trouble and can always save both herself and her friends. This was explained in the beginning of the movie when Ariel and Flounder are chased by a shark and she keeps on slowing it down. Plus how she managed to trap it in that... well whatever that thing was, proves she can save herself. Plus she's the first princess who actually saved her prince not once but twice, plus she killed the evil minions, not the actual villain but considered the heroines that came before her that was impressive. However she too does need saving sometimes. She needed her father to save her when the sunsets and during the climax by Eric. But overall Ariel wins this too.
Saving the day
Neither one of these girls saves the day but I believe they both had the potential to do it. Both caused the mess and many believed that they should be the ones who had to fix it all. Merida's mother saved the day and I think that's the main reason Disney and fairy-tales usually kill off the mothers. They're trying to teach kids to be able to take care of themselves and stand on their own without their mothers. Besides this was the first Pixar protagonist and they needed to leave an impression on us and having Merida save the way would've been a great way to do that. Ariel also doesn't save the day but she kind of had more of a reason. Eric had to do something or no one would've even remembered him so Ariel's reason kind of is better. Although you could've argued that they should've had both defeat Ursula. However overall Ariel wins.
Deals with their witch
Ariel's reasons for going to Ursula was to get a potion to turn herself into a human so she could live her dream and also be with Eric. Merida's reasons for making a deal with the witch was to change her mom. Merida didn't know what she was getting herself into and just kept on being all like "Are you sure this will work?" unlike Ariel who was quite skeptical. Ariel thought about the consequences and decided to take a chance. With Ariel it was more bravery and with Merida it was nativity. Plus Ariel's reasons are better. It was a big mistake for both of them though.
I believe both of them realize all the trouble they've caused but in different ways. Ariel at first says "Daddy I'm sorry, I didn't mean to, I didn't know, I..." but you can argue that could have just been the typical teenage girl getting caught. However you can see Ariel really feels horrible after her father takes her place by the expressions and the way she says "Daddy...". Also during the chaos when Ursula takes over it's kind of unrealistic that she wouldn't learn something or realize what she's done. However I guess her redemption was more in expression than in words. Merida however, even though several times she says it's not her fault, later admits it's all her fault. So for this round Merida wins.
Relationship with parents
Both have a really touching relationship with their parents. Merida and Elinor is very unique because for once it's the mother and daughter, though they kind of go over the top with it. With Ariel and Triton it's not the main focus of the story but is still very touching at the same time. With the parent relationship less is more and I enjoy all the touching moments with them. Merida and Elinor have some good scenes, like the scene where they're getting fish. But I didn't find it sad when it seemed like Elinor wouldn't change back, but I blame that mostly on how predictable, bland, and undeveloped the movie is. Plus their relationship seems a bit forced, while Ariel and Triton's is more natural and subtle. Plus the "I love you daddy" line is a moment that none of the Merida and Elinor moments can compete with. Plus Ariel and Triton actually had an impact on the life of a solider who was shipped out and had a strained relationship with his daughter. After he saw The Little Mermaid he called his daughter and they worked out their issues. This happened not too long after the movie came out and nothing like this has happened with Brave that they'd actually write a letter to Disney about it. So that definitely wins Ariel and Triton some MEGA points so Ariel wins this round.
OVERALL WINNER: Ariel
Ariel: Don't mess with the master Merida!