When you say "I like to play" it means that you like to do something, I like to cook, I like to run, etc. When you say "I like playing" you are saying that you like the actual doing of the activity itself. That might seem weird, sorry if it does. I am not a grammatist. Maybe this will help...... You can't say "I enjoy to run." You CAN say "I enjoy running." (I enjoy the activity of running)