Gym Shoes are high-top and have thick rubber soles with optional air pockets. They are bright in color and have a bold symbol along the side. Most importantly, they are meant to be worn on the bball court (or the squash court, depending on your level of swagger...just kidding).
Sneakers are classic, low-top, not best "fresh-out-da-box", and neutral in color. Think neutral Converse Chuck Taylors or Jack Purcells, or Adidas Sambas, or (my favorites) low-top Common Projects. These shoes work just as well with jeans as they do with khakis, and they can even be worn with a (slim) suit...again, depending on your level of swagger.
As a small side-note...as a former college basketball player I have a special place in my heart for a pair of retro Jordans...so if you are Kanye West, or have a similar level of personal style, and you can rock them with designer fashion, by all means, do your thang.
Yours in style, Style Blogger.