I have seen The Amazing Spider-Man twice now, and came up with reasons as to why it's better than Spider-Man (2002). Take it easy on me, its my first article.
1. Intelligence: TASM actually shows us Peter Parker's intelligence, rather than just telling us. In TASM, Peter builds his Mechanical Web-Shooters. Spider-Man (2002) simply had Peter use Organic webbing.
2. Spider-Man's personality: In Spider-Man (2002), Spider-Man did not taunt or make fun of criminals. In TASM, he does that. He was the Spider-Man that we know from the comics.
3. Love Interest: In Spider-Man (2002), Kirsten Dunst played Mary Jane, and was a horrible acting choice. She was just the damsel in distress who consonantly yelled at Peter. In TASM, Emma Stone plays an attractive, smart Gwen Stacy, whose intelligence rivaled Peter's own.
4. Realism: TASM was set in a realistic world.
5. Relating to the audience: In TASM, Peter can relate to many teens.
The Amazing Spider-Man 2 hits theaters May 2, 2014.