[***Possible spoilers for various storylines featuring Superman, The Fantastic Four, Wolverine, and several other characters.]
There are two primary schools of thought on superhero attire: The first being that superheroes should always have costumes when engaging in their adventures. While the other way of thinking is that costumes are fine, but so are everyday clothes, so long as they have the superhero’s logo or weapon on their person. Yes, there is a third school of thought where the practicality of function is the only thing that matters, but it is arguably not as prominent as the first two (The movie X-Men  is an example of this third school).
Now, the examples we are looking at for the first, standard school are Peter Parker (Spider-Man), Batman, and the Fantastic Four. While Jack Knight (Starman), Conner Kent/Kon-El (Superboy), James Howlett/Logan (Wolverine) are the examples of the second school of thought. There are also the “wild cards” in the form of Clark Kent (Superman), Princess Diana/Diana Prince (Wonder Woman), and Doctor Henry Pym (Ant-Man, Giant-Man, etc.).
In the case of costumes, one can easily point out reasons why characters would need them. For example, a lot of Parker’s gallery of villains went after him when his identity was publicized (Friendly Neighborhood Spider-Man issues 11 through 16 [2006 and 2007]). In the case of Batman, the costume is a form of help in fighting crime via intimidation and his tool-filled utility belt (Batman: Zero Hour ). The Fantastic Four, however, only seem to have been given costumes since people expected it during the Silver Age of Comics (Essential Fantastic Four Volume 1 ). Each example respectively represents: safety from revenge and anonymity; theatricality and practicality; and conforming to genre trappings.
The examples of the second school of thought represent some different things, but also a few of the same things. In the case of Jack Knight, we see him fashion a makeshift costume from a leather jacket, goggles, and a toy sheriff’s badge (Starman [1994-2001]). These items represent practicality and functionally, with a touch of theatricality. However, a good number of his adventures do not see him using these items, yet he still falls into a casual role due to his cosmic staff.
Meanwhile, Wolverine arguably looks more believable when he is in everyday clothes due to having originated as a loner anti-hero type. Since he is a loner anti-hero, he is representative of both rejecting costumes (wearing casual clothes) and wishing for inclusion as part of a group (wearing a costume)(Wolverine: The Brotherhood  and Giant Size X-Men #1 ).
The last of this second school of thought is the clone of Superman. Superboy, alias Conner Kent, has since his first appearance maintained a casual fashion sense, yet always with an S-Shield on display to represent the legacy that he is a part of (The Adventures Of Superman #500  and Superboy: The Boy Of Steel ).
Cover Art by Dan Jurgens and Joe Rubinstein, Cover Colors by Patrick Martin Superman #123 (1997)
Finally there are the “wild cards” whose costumes cannot change without greatly affecting a character’s recognizability. In the case of Superman, his outfit (including the red trunks) is iconic and removing a section or changing his outfit entirely makes him look goofy (Action Comics #1 , Superman #1 , and Superman #123 ).
While Wonder Woman, like Batman, should have options for different environments, her basic costume is arguably as iconic as Superman’s, and thus it is hard to change its design elements (Sensation Comics #1  and Wonder Woman #179 ).
On the other hand, Hank Pym is a “wild card” since he is too versatile, due to having so many heroic identities (Ant-Man Tales To Astonish #35 (1962) and Dr. Pym in West Coast Avengers #21  being two examples). In other words, “wild card” characters are either too iconic or too versatile, and therefore changing their looks only lasts for short periods of time.
In conclusion, these two primary schools of thought have merit to them, but deciding on what works best in terms of costume can only be determined in the context of each character.