Why is 'the' added before Wasp but not before Antman from the movie title "Antman and the Wasp"?
"Antman" and "the Wasp" are superhero masked identities, which you can treat as a sort of brand name, and should therefore be treated as proper nouns.
Proper nouns do not require an article, but sometimes include one. "The Wasp" is one of those which includes an article, so it has "the" except in the vocative. Antman is not one such, so no article is required.
There's no rule for which proper nouns include an article and which do not; it is determined by whoever decides upon the name.
So why is the Wasp sometimes addressed just as "Wasp", without the "the"? Well, there's a convention for personal pseudonyms containing "the" where the article is dropped in the vocative, and sometimes in other informal contexts (so she'd be addressed as "Wasp" but referred to formally as "the Wasp"). Okay then, why "the Wasp"? Shouldn't it be "The Wasp", with a capital T in "The"? Actually, no. Such names can be treated like titles (in the sense of titles of artworks or literature), for which the convention is that some short words, notably including "the", are not capitalised despite being considered part of a proper noun.
A common noun follows normal rules of grammar—such as being preceded by an article. On the other hand, a proper noun (name) normally does not, unless some kind of disambiguation is involved. (There are several people in a room with the same name.)
So, while you have the ant crossing the floor, you have Ant-Man crossing the floor.
But I was puzzled by something. Is the actual name of the character The Wasp or is it just Wasp?
If it's just Wasp, then no article should be used. Meanwhile, if it's actually The Wasp, then an article should be used but it should also be capitalized.
Oddly, public media seems to keep using an article in lowercase when referring to this character specifically.
Even more alarming, the official Marvel wiki is all over the place.
It defines an entry for the character under the name Wasp (no article). But it then goes on to describe the character in this way (text in bold is mine):
As Ant-Man and The Wasp, they defeated the monster and sent it to its own dimension, and had several more adventures together. To the Wasp's credit her first solo victory, despite the protests of her absent partner, was against the maniacal Magician.
Unfortunately, this leaves me not knowing what the title of the movie should be—because I can't determine what the name of this character actually is.
However, based purely on normal grammar and common styling, I would argue that the title is strange.
If her name is just Wasp, then the title of the movie would normally be:
Ant-Man and Wasp
If her name is The Wasp, then the title of the movie would normally be:
Ant-Man and The Wasp
But for some strange reason, there is a confusing mix of naming and style in this particular case. Arguably, if there is some reason for this unusual formatting, and media commonly does put a lowercase article in front of her actual name, then the title as given is "correct"—as far as public opinion goes.
Still, you are certainly right to be puzzled by this.