I can't speak for others, but when they have to replace the core-characters (such as Peter Parker/Spider-Man himself), I just have a very hard time to watch it, if it would take part in the same "universe".
Therefor I think a reboot is a good thing, because I would rather see a brand new story with new actors, then to watch Spider-Man 4 (continuing Sam Raimi's movies) but with some actors switched out.
There are cases where the directors/producers have replaced a main-character with someone else, and tried to continue like nothing happened, and it is very rarely a good thing.
I enjoyed Raimi's movies when I first saw them, but when I look back at them I feel so-so about them.
As movies they are pretty good, but being a huge Spidey-geek when I was younger (reading and collecting over 300 magazines), there were plenty of things in the movies that bothered me, minor details mostly, but things I thought they could improve.
A good example is the webshooters, for people who know the comics, they're a huge part of it, but Raimi decided to go another path with the built-in web, and while that might have been better, it was still a big thing in the comics that they screwed over in the movies.
So with that detail, I am glad that they will fix it in the new movie, at least judging by different pictures.
As for Spider-Man 3, I really want to like it, because it has my favorite Spidey enemy in it. (Venom)
However, apart from the obvious crapfest that was Peter going emo, I thought they really killed the character Venom, he didn't get much time in the movie, and the little time he got wasn't really that good.
It has obvious reasons since Raimi didn't want Venom at all, it was something they "had" to implement due to fans wanting it, but I'm still very disappointed about how it turned out.
If I could wish for anything, I would love the sequel to the new movie (it's not exactly confirmed but there will most likely be one or two more in this installment) to have Venom in it, as the main villain.
I just think the character could be so epic on screen if they made in a good way, gave him the proper amount of screentime and have his character develop during the movie.
Oh well, wishful thinking.