To me it's pretty simple: you should dress better than your audience. If the audience is mostly in jeans and sweatshirts, then a jacket and dress shirt with dress pants and shoes will do. If your audience is in blazers and dress shoes then you should wear a suit. If your audience is in suits/ties, then you should wear a tux. And if you have no idea which scenario you're facing then it's better to err on the formal side. 

Most of us have had occasion to dress up at one time or another. This is one of those times. Adding appropriate stage clothes to your range of sartorial options is nearly as critical as getting the right instrument to play. It's every bit a part of the whole picture. It's true that this type of clothing is relatively expensive, but they last a long time and, after all, it matters. 

I've seen many attempts at compromise. The student who wears a nice jacket and pants but with white gym socks and ratty loafers. Or a nice blazer with a wrinkly mis-matched shirt. Or a nice dress shirt with a crummy off-colored sweater. Everyone needs a decent set of dress clothes and preferably, eventually, a few different options. 

Why do we dress up to perform? Doing so is of course tradition; the audience expects it. If you come out on stage in jeans they'll think they walked in on a rehearsal. It is a sign of respect and intent: it signals that you take it seriously. Are there circumstances that don't require dressing up? Sure. Playing in cafés or bars is generally more casual. But my subject is what we do on the concert stage. 

Do women have different challenges than men? Naturally. But also additional opportunities. As they can choose to wear either pants or a dress/skirt their options are many. But if choosing to wear a dress or skirt, a long one is best. I saw a perfect demonstration of this at Oberlin one year. A male guitar student of mine and the female singer he was performing with decided, as a lark, to switch gender in their choice of dress: she wore a tux (she looked spectacular), and he wore a frilly blouse with a skirt and looked ridiculous. The audience laughed when they walked out on stage. But when they began to play, the true hilarity began. He evidently hadn't considered what playing on stage in a skirt would be like beforehand. In a room with an elevated stage like our Kulas Recital Hall, the audience's sight-lines are such that, when he put his left leg up on the footstool and threw his right leg over to the side, he flashed the audience a perfect view of his tighty-whiteys underneath. We burst into hysterics. At this point, he had to try and play. And, suddenly feeling coy, he tried to put his legs together and play, an adjustment which threw off his playing angles and he struggled to get through the material. But at that point, we truly didn't care. Watching him struggle to avoid flashing us again and still keep his poise for playing kept the audience in stitches throughout. I wish I could remember the singer. But the guitarist was the inimitable Justin Riberio.

There are many styles of dressing up for performing. Looks go in and out of fashion. Tuxes were universal for a long time. Regular suits, in dark colors took over. Then younger looks, like suits with dress shirts but no tie, or with T-shirts came into vogue. Now it is common for performers to dispense with the jacket altogether and just wear a black dress shirt. There was a time when guitarists commonly wore turtlenecks on stage.  All of these are fine. It's best to hew towards conservative looks on stage. After you've established yourself as a player worthy of attention for a while, pushing the clothing envelope some is fine. Pushing it at the outset can make it harder to be taken seriously.