Tough choice.
Both are very good phones, and are at the top of the list in sales.
The iPhone does have a better reliability record.
Samsung makes a very nice phone (my previous phone) but they use plastic for most of the phone, where the iPhone is aluminum and has a nice feel to it. It's really light as well.
Battery life is amazing on the iPhone. Apple really has it nailed down when it comes to power management.
As stated, the S3 will be replaced soon by the S4, But the iPhone will also be seeing a replacement this year. Not as soon as the S3 though.
If you can take the size, the Note2 is a nice option as well. I like it, but it feels like I have a poptart in my hand when holding it . It could be handy though if you don't have a tablet, and want to use your phone as a tablet as well.
Really though, try all the phones out at the Vzw store that interest you. One good thing with Verzion. They have a 30 day return policy. If you buy the iPhone and don't care for it, take it back and pickup the S3 or vice versa. They will charge a restock fee though. Still not bad though.
What do I have?? An iPhone 5. What did I replace with it? A Galaxy Nexus (still have it).
Goodluck. Really you won't be disappointed in either phone.