Punisher vs. The Spider-man
Is the word "the" used however we feel like? Could I call them "Punisher" and "The Spider-man"?
Spiderman is not an ordinary word while punisher is. People will know that a punisher is someone who punishes, but what in the world is a spiderman? The use of the with punisher then suggests that in the movie a particular character is being referred to, while Spiderman needs no such addition. In the same way we say Catwoman and The Joker.
However, Storm is not called The Storm because she is not really a storm, Superman is not called The Superman because he is not really a superman, and Cyclops is not called The Cyclops because he is not really cyclops. In contrast, The Punisher really is a punisher and The Joker really is a joker, in the ordinary sense of the words punisher and joker, much more so than Storm, Superman and Cyclops.
The reason "The Punisher" is written that way is because the is part of the name. The reason Spider-Man is written without the is because it is not part of the name.
Proper nouns are special, because a proper noun is a reference that points a single entity.
Articles (a, the) in English are most commonly used in discourse to help disambiguate nouns. When I say "a duck", it usually means that I am not referring to any duck in particular. When I say "the duck", it usually means that I am talking about a duck that is known to me and to the listener(s), either because I have talked about it before, or because there is only one around.
Since proper nouns already point to a single unique entity, there is no need to add an article.
However, if the or a is part of the name, then it should always be used with the name, and never change between a and the based on context. Thus, I always say "A Clockwork Orange" to refer to the book, and I always say "The Edge" to refer to the musician. Those articles are not part of the discourse information, they are simply part of the name.
So, "The Punisher" is always "The Punisher" and "Spider-Man" is always "Spider-Man".
(There are certain exceptions to this, but they are very special cases. For example, there is the company called Wal-Mart, which should not take any article, unless you aren't talking about the company as a whole — if you are just talking about one of the stores, you could say "I'm going to the Wal-Mart" or "I'm looking for a Wal-Mart in town".)