There are a ton of shows that fall in to this for me. When I was in college me and my roommates pretty much only used Netflix because HD was unavailable at our apartment complex. Live sports were basically the only thing we used our cable for and even then if it was on ESPN we watched in on my 360.
Some of the notable shows I watched streaming between Netflix and Amazon Prime that I hadn't seen before are Arrested Development, Firefly, Freaks and Geeks, Trailer Park Boys, Party Down, and I've gotten part way through Fringe over the last couple months.
There are also lots of shows I missed in the first couple years and then caught up so I could watch them when they come out on TV. Some of these are Breaking Bad, Mad Men, Dexter, Weeds, Spartacus, Downton Abbey, and American Horror Story.
I think it really helps a show's success to be available on Netflix. I know that many of the studios think it will hurt their disk sales or live viewership numbers but I think it has the opposite effect. I'm not someone who will jump into a series after a couple seasons without knowing what is going on. If I have missed the beginning of a show I will usually just pass even if I hear good things. If you give me an easy way to marathon your past seasons without commercials I will be much more likely to catch up in time to be one of your weekly viewers.
I highly recommend Party Down and Trailer Park Boys if you haven't seen them. They are two of my favorite comedy shows and I never would have seen them if it weren't for Netflix.