～にいく／～にくる (where ～ is the -masu stem of a verb) and ～ていく／～てくる are both standard constructions, and there is some overlap in their usage.
If you say ～にいく then it implies you are going to a specific place in order to do ～. For example, 公園に遊びにいく means "I am going to the park to play". So in your case, 今度遊びに来てください means "Please come (here) to play next time", i.e. it's an invitation to come back to play with the speaker.
On the other hand, ～ていく has a few different interpretations depending on context, but one main interpretation is that the subject is going somewhere, and doing ～ before or as part of the journey. For example, 歩いていきます means you're walking somewhere (you're going there, and walking is part of the journey). ～てくる is similar, but it implies that the end of the trip involves coming back home or to coming to the person being addressed. For example, when you leave home you will say 行ってきます, meaning (roughly) "I'm going out, and will be coming back". You could also say だいどころからいすをとってきます to mean you're going to bring a chair from the kitchen.
In that context, 遊んできてください sounds like an invitation to go away, play, and then come back, which sounds more like what a frustrated parent might tell their hyperactive child.