The only difference seems to be that selecting the "inn" option wastes money. Is there an advantage? There doesn't seem to be.

Using the Inn get to ya place faster than campfire.

Very slightly faster and only really if you're travelling a long distance. The only real use for it is if you're working under a time limit (for a quest or the like) and that spare day or a few could make all the difference.

It also seems using inns gets you there more rested. When I've camped out I got to the place with somewhat less fatigue.

The only reason I'd suggest "safe" travel is if you're going to Daggerfall; if you travel recklessly and end up there at night, you're going to get some dude VEEEEEEENGEAGEing all up in your shizz.

If I recall, in any walled city, if you arrive there at night you can also be attacked by a group of rogues/thieves/assassins at the city gates.
But I don't remember if it happens when arriving at a walled city at night, of it happens when you're released from jail at night.

I just remember being butchered by thugs when I got out of jail. It's even more dangerous when you get released from jail with 1 HP.

Yes, the game is great at kicking you when you're vulnerable. I'm still terrified of the game's arrow sound effect. In those situations, at night, you're usually dead before you can find the source of the arrows.

Me: "Uh oh, no HP and the gates are locked. Crap, I haven't saved in a while. I sure hope there aren't any rogues around with bows and..."

Arrow: "Thwack!"

Crow: "Caw! Caw!"

And in case you would decide to play as a vampire: travelling recklessly can have you entering a town during the day.

Vampires and sunlight don't match.