From what I've seen, Reboot and Remake are nearly interchangeable.
No.... completely different gameplay. Different levels, different weapon design... if you call DOOM 3 a remake of DOOM 1... you must have an IQ below 100. DOOM 3 was a REBOOT. There's a difference. If it was a remake it would have been much more well recieved and much more like the original DOOM 1 that all of us have grown up with.
Doom 3 may of seemed like a Reboot because it had different art direction and things, but the thing is, it was a remake because the Story was pretty much the same, most of the monsters reappeared, and it kinda of had the same idea. (Start in Tech Base, Tech Base becomes fleshy, Hell [Then Back to Tech Base, then suddenly to Dig Site, and then back to Hell again, but anyways]).
Reboot is usually trying to kickstart a franchise again, but at the same time, when ever someone does this, they're usually remaking something old, giving it the same name and story, and either changing some things to make it fresh, but otherwise it just has some new polish on it. RoboCop for instance, or Total Recall maybe (Not sure, as I can't really qualify this as a remake of the movie, but just another Adaption of 'We Can Remember it for You Wholesale'.)
Something like that anyways.