These days, it's largely true that Wolverine has little need to hide is identity. But that wasn't necessarily always the case.
When Wolverine was first introduced in Hulk 180/181 (he technically showed up in one panel of #180), he was an agent of the Canadian Government - a post where a secret identity could be quite useful. This usefulness continued as he was then recruited by Department H to be a part of Alpha Flight.
From here, Wolverine would join the X-Men. It is however, disingenuous to say that the X-Men didn't really wear masks at that point. In fact, most of them did (or they had alternate forms which effectively acted as masks).
Over time, that's certainly changed - but as the X-Men were frequently (i.e. pretty much constantly) branded outlaws, it's never a bad idea to keep your identity secret in an attempt to avoid hassle if out in public incognito.
There are way too many members of the X-Men to consider if you want to look at the entirety of the team roster, but going over some of the more prominent/infamous members:
- Cyclops: Usually wears a mask in most versions of his costume
- Jean Grey: Wore a mask in early costumes as Marvel Girl
- Beast: Wore a mask prior to gaining his blue fur.
- Iceman: No mask, but does not need one due to his ice form
- Angel/Archangel: Originally wore a mask until he was outed as a mutant. After being rebuilt, Archangel was blue skinned - effectively a mask versus Warren's normal skin.
- Wolverine: Mask
- Storm: No mask
- Colossus: No mask, but does not need one due to armored form
- Nightcrawler: No mask
- Havok: Mask in early costumes
- Polaris: No mask
- Kitty Pryde: Wore a mask in early costumes
- Psylocke: Wore a mask with most costumes prior to gaining new body
- Forge: No mask
- Gambit: No mask
- Jubilee: No mask
- Rogue: No mask (of note: she started as a villain and her identity was previously known)
- Bishop: No mask (of note: he's from the future)
If you start looking at the list, you'll note that many of the older characters wore masks, and it's the newer ones who don't. This trend largely started towards the end of the Chris Claremont run.
Additionally, Wolverine has taken on disguises in the past. Most notably as "Patch" in Madripoor. While he rarely (if ever) used his trademark mask during these days, he would often wear an eyepatch in an (poor) attempt to hide his identity.
A better question would be to ask why so many of the other X-Men don't hide their identities.