Wolverine's calling card has always been that he's the most mysterious of superheroes. He's been around for a long time, but his age is unclear because of his healing ability. He was experimented on and had adamantium grafted onto his bones, but can't recall who did it. Probably the only thing that's known about him is that his name is Logan — but is that even true? In the new Wolverine film, which is called Logan, he also goes by the name "James Howlett." So is James Howlett Wolverine's real name?
It actually is. While the character is commonly known as Logan, including throughout the majority of his comic book history, comic book histories are often changed. Comic book characters are frequently subject to "retcons," where a new writer will change something about a character's history and make that the new canon. This happened in regards to Wolverine's name back in 2001 in the limited series Wolverine Origins. Since the character's creation in the '70s, he knew little about his past, and went by the name "Logan" — no last name — when he wasn't using his codename Wolverine. But this series finally revealed the character's past, including his birth name of James Howlett.
In that story, James Howlett is born in the 1880s in Canada to John and Elizabeth Howlett. However, he eventually learns that his real father was a man named Thomas Logan, who kills John Howlett. Young James's powers manifest upon seeing his dead father, and he kills Thomas with his claws. He flees to the countryside after committing his crime, and starts going by the name "Logan." That's the name that sticks with him after he loses his memory after participating in the Weapon X program.
The story of James Howlett's childhood is actually canon in the X-Men movies, too, though I'd forgive you if you forgot about it. It's depicted in the 2009 film X-Men Origins: Wolverine, which was at least in part based upon the 2001 comic. While most of that movie is no longer considered canon thanks to subsequent movie retcons (see: Deadpool), it seems that Logan's backstory as James Howlett remains intact even in the new film.
While Wolverine's real name may be James Howlett, to many fans he'll always be known as Logan. I guess that's why his swan song movie is called Logan instead of James Howlett.