The conductor is like the director of a movie. The actors technically do all of the work. They each have their own lines just like each musician has their part to play. Very often though, they are focused only on their part. And as performers, if left to their own devices they would probably try to out-perform the people around them. The conductor keeps that in check. It’s his or her job to make the whole production enjoyable, not just the individual performances.
The conductor brings the performance together by making sure everybody stays on track and tells some people to play louder or others to play softer.
Playing a single instrument in a band doesn’t actually give you a good representation of what the band sounds like to the audience. If you are a clarinet, for example, you can hear the clarinets REALLY well, the flutes pretty well, and the brass sitting behind you will be particularly loud. It can be difficult hearing the trombones though on the other side or something like that. While good players do know how to balance and how to listen, it helps having the conductor signal “more, more!” or “back off a little” since they are hearing everybody.
They also cues different sections which is helpful for placement issues, as well as when you get into mixed meters such as 9/8 where it can be broken up as 2+3+2+2 and the next bar be 3+2+2+2 and keep switching around. In cases like that it is helpful having somebody denoting where the emphases are.
And then there are cues that come down to the conductor. The conductor decides when a certain instrument comes in. For example the brass section is holding a long note, and then the conductor decides the flutes to come in, and then the clarinets. The conductor also decides how long pauses last and whatnot. Where it is difficult to do with a large band.
The conductor’s real job is during rehearsal. They practice a lot, and I mean a lot to get the performance down. During that time, the conductor is a lot more engaged. She will start and stop the music at different parts, give verbal instructions, and nail down every little detail. When the symphony actually gets around to performing for the audience, yes, they are all just reading their sheet music and not paying as much attention to her waving the wand. Technically, they could probably get along fine without her there at that point, but they would not be able to get to that point without the conductor during rehearsals.