"He plays on his mobile phone" or "He plays games on his mobile phone"?

Both are correct grammatically speaking; however, one might "play around" on his/her phone without actually playing games. So if you wish to specify exactly what the person does, you would want to say: "He plays games on his mobile phone."

In a loose sense, no matter what the person is doing, you could say "He's playing around on his phone," or "He's messing around on his phone." That would not necessarily mean that "he" is playing games, and the question could be asked: "What exactly is he doing on his phone?" Then you might respond with: "He's playing games on his phone."