Ariel not signing a contract: In this version, Ariel plucks a scale from her tail to initiate a deal with Ursula. This is done to avoid what many consider a plot hole of the original film - if Ariel can read/write, she could have just written a letter to Eric explaining that she was indeed the woman who rescued him, which he ruled out when he met her as a human because of her muteness, as his rescuer had been singing.
This change is also why Ursula is able to toss in some memory loss to the spell, it's not a contract so she doesn't need to be honest about the effects or stick to the agreed terms. As long as Ariel gives her the blood and scale she's free to do whatever else she wants with the spell.
Of course mermaids wouldn't have tears; their main biological function is to keep the eyes clean and hydrated, which is obviously unnecessary for a being that lives under the sea. Even if they still were biologically capable of producing tears (being half-human and all), it would only happen as an emotional response, and, again, tears are basically saline water, so they'd be unnoticed.
Prince Eric being given a new backstory: As has been pointed out in critiques of the original film, Eric is a cypher, that is, a character that virtually nothing is known about, so we don't know what Ariel saw in him (Love at First Sight notwithstanding). In the remake, they addressed the "problem" by establishing a larger backstory for him: sea orphan adopted by the royal couple, does not always agree with his adoptive mother, etc. He is also given a curiosity trait similar to Ariel where he collects things found on the ocean (he has his own "grotto"), and wonders how it's like down there, something the movie used to bond them together.
Ursula protests that she's not half the monster Triton thinks she is. No, she's absolutely 100% the monster he thinks she is. Or even worse than that.
Ariel eating the tropical flower she was offered makes sense because the other vendors offered her a drink and food to sample, and she had no reason to assume the flower wasn't edible as well. She's also shown horrified at the realization that humans eat fish, which indicates that mermaids are vegetarians - or at least her family is.
Some viewers have noted that, unlike the animated version, Vanessa doesn’t even bear a passing resemblance to Ariel. She doesn’t need to. Ariel, who (obviously) “resembles” Eric’s savior, was immediately dismissed as Eric’s mysterious girl because she was voiceless. Vanessa both has the voice (the clincher) and the Mind Control magic to stop Eric from realizing the ludicrousness of mistaking a Black redheaded mermaid with locs for a white woman with wavy brown hair. (Even then, she doesn’t succeed, as Eric starts questioning it the moment he’s away from her, asking after Ariel and telling Grimsby he doesn’t understand why he needs to marry Vanessa.) Ursula took advantage of Ariel’s accomplishments to further her own goals.
Much has been made about the Colorblind Casting of Triton's daughters, which can be chalked down to merfolk genetics being different than humans - their entire society is different, there's no point in assuming that human concepts of "race" and "ethnicity" should apply to how a mermaid should look. Even taking this into account, assuming adoption wasn't in play, siblings Triton and Ursula are of entirely different species - he's a merman, she's half-octopus -, and their respective actors have different ethnic heritages. However, in Classical Mythology, Poseidon's children are of vastly different species: cyclops, sea-horses, giants, and more. With his consort Amphitrite, he had a son named Triton, who is often portrayed as a merman.
The movie makes it clear that Ariel's singing isn't just a talent but a specific kind of mermaid "gift" that can lure sailors to their doom and even revive them from drowning. This creates an issue: Is Eric's love for Ariel genuine or just an effect of her literal siren song? Luckily, due to Ursula's spell making Ariel unable to remember that she has to kiss the prince, and Eric getting to know Ariel over time and bonding, it doesn't matter if her song initially hypnotises him because he still chooses her as a mute girl even after Ursula uses Ariel's voice to lure him later on. The live-action establishes on both parts that Ariel and Eric's affection for each other is truly genuine, not a result of having an agenda (wanting to be human and needing a kiss) or magic (the siren song).
It's buried in the chanting, but Ursula really did tweak the spell to make Ariel forget about the kissing clause. Where the animated version sings "Larynxes, glossitis" note "voice box", and "inflammation of the tongue", the live-action version sings "Amnesia, glossitis". note "memory loss"
The dippy Scuttle confuses Ariel's mission to kiss Prince Eric for killing him, which annoys Sebastian... but, really, she's '''not''' too far off, if one knows the end of original Hans Christian Anderson tale. Part of the young girl's cursed bargain with the sea-witch, to get back her fins and live as a mermaid again, is to stab the prince in the heart as he sleeps and let his blood wash over her feet! (She doesn't do it, as she loves him too much, and opts for a noble sacrifice instead.)
Ursula keeps going on about how, should Ariel fail her quest, she would belong to the Sea Witch. What does she want with Ariel? What would have happened had Ariel been unable to defeat her and revive Triton? If Triton had refused to cede his power? If the seas refused to accept Ursula as Triton's successor to the throne (given that she took it by force), would Ariel have been turned into a puppet queen - at the tender mercy of someone who can manipulate minds, no less?
Ariel should be lucky her father so willingly gives up the trident to save her. We see that rather than turning them into polyps like in the original, Ursula outright MURDERS the victims who fail her tests. Given she's after Triton's crown and power, had he refused to give it up, she likely would have kept Ariel alive to torture her day in and day out until he eventually broke to save his daughter. While Triton's death was relatively quick, Ariel likely would have had a long, painful, torturous existence ahead of her.
In the original fairytale, the mermaid not only gives up her voice for a chance to be with the prince, but also her extremely long life. Because mermaids don't have souls, they can live for hundreds of years. In this version, King Triton is not only explicitly referred to as the Sea God, but the "Guide To Merfolk" tie-in book says he has been around as long as humans can remember. Ariel, either by virtue of being the daughter of a Sea God, or due to merpeople being naturally long lived, is likely damn near immortal as a mermaid. By becoming human, Ariel didn't just give up her family and undersea friends. She's also significantly reduced her potential lifespan from thousands, if not millions of years...to a few decades.
