I spent a lot of time researching these two options. I had a slightly different short list of functionality that I wanted (both that I could or couldn't live without). The G2 has better hardware specs than the S4, as you probably know, and the benchmarks generally beat it (which gets to your multimedia/gaming requirements). But, they're both great phones and i'm sure you wouldn't have problems with either. They are similar, outstanding phones.
I decided on the G2 because of its camera OIS (only android phone that has it, period). And, it consistently got better reviews on almost all photography-type sites than any other android phone (s4 included). I love the camera, it is very good (for a phone). I just put its HDR pictures and videos on a 720p 50 inch TV - they look outstanding. Obviously not as good as a real camera...