I must admit, I have looked at this question for some time, and have spent a long time thinking about how to answer it. I have a few ideas now that I would like to share. Firstly, you need to remember that this play is about two parallel worlds - the world of the forest and of Titania and Oberon, and then the world of the city and Athens. In a sense you could argue that the setting here is used symbolically to represent order and reason (Athens) and disorder and chaos (the woods).

Given that we have these two parallel worlds, we also have two types of underclass that are there to pander to the whims of the nobility. The fairies are clearly at the beck and call of Titania and Oberon and the Mechanicals are under the power of Theseus and his soon-to-be-queen, Titania. Thus one comparison is the social standing that each group has - both are forced to please their superiors, and both groups also are a source of fun and comedy in the play.

You might find it productive as well to think about how Puck and Bottom are presented and what parallels can be detected. Both it seems are a force of disruption in the play - with Bottom determined to take all of the roles when we are first introduced to the Mechanicals, and Puck (wilfully?) mistaking Oberon's commands and anointing the eyes of the wrong Athenians.

I hope these ideas help you think through some of the comparisons - I must admit, I think there are more contrasts than similarities, so it might be worth going back over the text and exploring this issue further. Good luck!