Rock climbing is possible in sneakers, and even barefoot for that matter, but it is highly recommended to climb in climbing shoes as they will provide a better base for standing on small climbing holds, and give you better stability and grip on the wall.
Sneaker tend to have very soft rubber for their sole, and a generally inefficient shape and size when it comes to climbing. Due to this, trying to stand on small foot holds may be impossible since sneakers have such a large and soft sole, your feet would just slide right off of them.
Climbing shoes on the other hand have much harder rubber, the shoes are much tighter, and they are much less bulky. They are very thin in comparison to sneakers. All of these attributes, plus their shape, give climbers the best performance possible when it comes to feet work.
In climbing shoes, you can actually feel what you are standing on, this allows you to trust your feet much more than sneakers, which is something that for many is very difficult, but crucial.
