I really don't feel like I can compare the franchises yet, since the new one only has one movie so far, and the other one has three. However, just going by first movie vs first movie, I feel like Raimi gets a slight edge.
Raimi is without a doubt one of my favorite directors, and he knows how to put together a movie. Raimi decided to go for a more oldschool Spidey, and Webb went for a more modern approach. I like that. I like the fact that both versions are different. For me there are things that works and doesn't work in both movies.
If we start with TASM, I love the decision to go for the webshooters. It's closer to the comic, and I think Andrew Garfield is amazing as Spider-Man, with the witty banter while fighting criminals. If there was something I missed with Tobey as Spider-Man, it was that he didn't really do the funny lines while fighting, which is a bit of a trademark for the character. I like it that Oscorp gets a center role, and it feels more natural. Andrew and Emma has really good chemistry, but I also feel like Tobey and Kirsten had that as well, so maybe a tie on that part.
What makes me lean toward Raimi's first movie over TASM1, is just how great and memorable it was. There are several iconic and really good scenes, from the Spidey kiss with MJ, to the part when Peter sees Uncle Ben dying, and he overhears the police radio, goes through the alley and switching to his costume. That's really good filmmaking. I also enjoy that the "getting used to the powers"-part lasts a bit longer in Raimi's movie. Uncle Ben dying also had more impact in Raimi's movie. "With great power comes great responsibility" was a much more memorable moment, than the last words said by Uncle Ben in the Webb version. I also liked the score better in the original films, the theme is amazing, and one of the best overall I'd say. Can't even remember the new theme.
I don't feel like TASM1 had as much memorable stuff in it. Not to say it was bad or anything, but few scenes were amazing to the point of feeling iconic. I feel like Webb did a fine job for his first action movie (and second movie overall), but you rarely hit a homerun on the first try. I feel like TASM2 has the potential to be the best Spidey movie so far, and I think Webb has grown a lot.