Firstly I would like to say both shows are great, they carry a very strong point to each scene and the acting is very good. Some actors are better than others, but there are things you must consider. For starters, Amy Poehler is almost a veteran in the game. She's already done SNL and so many movies, its a no brainer why her show is edgy yet fun while still making it decent enough for young and old people.
Then there is Lena Dunham, who is younger but has way more leverage than Amy. Her show 'Girls' is seen on a network different from Comedy Central (I think). Therefore her character and actions are much more dramatic and intense. The most important thing about both shows is that they both tell stories about women who are in their 20's.
The question I ask is which one are you? In HBO's Girls, you're a 20 something with a a fine ass dude living with you in a decent part of New York where you have a career and 3 other friends who are trying to determine if you're worthy of an Instagram photo. In Broad City, you're a 20 something pot head with the world at your fingertips living in a poor part of New York with strangers. You work a low end job and you know what parkour is. Same city different taste.