When I first discovered the character, in comics, Hugh Jackman wasn't yet associated with him, and he was allowed to be kind of squat and ugly. He's definitely been prettied up now, though I'll give Jackman credit for understanding the character psychologically and mastering some of the iconic poses.
Logan is probably my favorite superhero character, besides those I myself have written. He's an antihero, in that he's not always interested in being a hero and he's willing to use deadly force. I like him because he's got such a relentlessly tough and grumpy exterior, but he's got a marshmallow of a heart when it comes to people who need protecting.
He has an especial soft spot for young women and in many of his storylines has had a young protégé: Kitty Pride, Jubilee, Rogue, X-23, Ms. Marvel (Kamala Khan). I've admired the relationships for avoiding sexual complications (and the ick factor of teenager with an old dude) and for the way he manages to protect these women while still helping them fight for themselves. He's egalitarian in his own way, and old fashioned and modern at the same time, teaching that the best defense is a strong offense.
He's an amazing character, with a healing factor (at least in most storylines) that slows his aging and gives writers a long history to play around in. He's been a soldier, a mercenary, a recluse, and a hero. I'll keep reading him as long as they keep writing him.
Don't forget to check out my own superhero stories. Change of Life, book two of my Menopausal Superheroes series just came out a few days ago!