If you were to only read "Amazing Spider-Man", you would miss out on the following major changes/explanations/plots:
Spidey got his organic webbing in "Spectacular".
Harry Osborn died in "Spectacular".
Doctor Octopus died in "Spectacular".
Peter handed the webs over to Ben in "Spectacular".
Ben Reilly appeared and kickstarted the Clone Saga in "Spectacular".
Ben was "revealed" as the real Peter in "Spectacular".
Kaine "died" in "Spectacular".
Ben Reilly's Spider-Man costume and blonde look premiered in "Sensational".
Peter's powers return in "Spectacular".
Norman Osborn returned, Ben Reilly died, Peter was "proved" to be the real deal, and the Clone Saga ended in "Spider-Man".
Hobgoblin retired in "Spectacular".
Aunt May was revealed to be alive in "Spider-Man".
Flash Thompson was paralyzed in "Spider-Man".
Eddie Brock sold the Venom suit to Mac Gargan in "MK: Spider-Man".
The Sin Eater Saga, which had Daredevil finding out who Spidey is and acted as a very important stepping stone to creating Venom, happened in "Spectacular".
Eddie Brock was revealed to have cancer in "Spectacular".
Spidey gets the black suit in "Secret Wars"
Ned Leeds is killed in "Wolverine vs. Spider-Man"
Doc Ock's funeral is in "Funeral For an Octopus"
Roderick Kingsley is revealed to be the original Hobgoblin in "Hobgoblin Lives"
Norman Osborn is outed and arrested in "The Pulse"
Peter loses his powers in "The Final Adventure"
Spidey joins the Avengers in "New Avengers".
...Just to name a few.