Let's play with each other
Let's play together
mean the same thing?
Do they have different connotations? What's the difference?
I feel that on the literal surface, the two phrases have very similar meanings since they both mean doing something with the other person/people. However, "play with each other" has a strong sexual connotation, because when you play with something, the thing you play with is your toy. So, when you play with someone, there are two meanings: playing together with someone, or using a person as a plaything. When you put those two meanings together you get the obvious sexual connotation.
Thus, playing together avoids any hint of impropriety.
Note: other similar phrases:
Of course, all of this rests on the fact that "playing" has a connotation (among adults) of sexual activity (or it's an outright euphemism). So you can reach the same connotation even among adults if you say "let's play together"... people have to be clear ahead of time what is meant by play.
“Playing with each other” does not have a connotation of collaborative play: it’s more like I play with you, you play with me. It can even have a tone of playful teasing. Not to mention, searching for “playing with each other” on Google brings a lot of porn results (links not provided, do your own research work!).
If you talk about kids at a playground or playdate, or about adults playing board or card games, then “play together” is definitely more appropriate.