Let's do a species to species comparison for Bottom and Puck. Bottom is a human and Puck is a fairy. As a human, Bottom is not excessively handsome (although he thinks he is), he's not great with language, and he's a bit coarse and grating to get along with. While Puck is much better with language and good looks than Bottom, he is not that way when compared to other fairies. He is called a "hobgoblin" by another fairy in act 2, scene 1. That's not exactly a great compliment.
I would say that another similarity between Bottom and Puck is that they both have a self inflated and bloated opinion of themselves. Both characters are supremely confident about themselves and enjoy bragging about it and being overbearing most of the time.
You might think this next one is a stretch, but I'll give the example anyway. Bottom is a weaver. He makes stuff by weaving various fibers together. Puck is not a weaver of cloth, but he is a weaver of lives/love/mischief/etc. Without Puck and his magic, the entire play would never have gotten off of the ground.