A few things to consider...
The quick and dirty answer is: Get the S4.
For a more complex answer, here's what you have to consider:
1. Price. If you're on a budget (or simply want to save some $), the S3 will be considerably cheaper. I think the S3 is going on Amazon right now for $49.99 (and may even be cheaper other places), and the pre-order price for the S4 is $199 (after mail-in rebate).
2. Longevity. No, the S4 may not be a *huge* upgrade over the S3, but don't think that the improvements are insignificant. They both may have the same amount of RAM, but the S4 processor is better. In terms of updates, the S3 will probably still get at least one more major update (no guarantees, though likely), but the S4 will be supported longer. It is important to remember that if you plan on sticking with Verizon's update schedule and buying subsidized phones, you're making a 2-year investment here, so by the time your next update comes around, the S3 will already be 3 years old. (so it could be the equivalent of owning a Galaxy S1 or HTC EVO 4G right now)
If you are not a gamer, the S4 16gb should be fine. If your are a gamer and already have a 32gb S3, best to stick with that u until a 32gb is available.
I do have one and will wait (perhaps futile).
3. Storage. From the evidence available, the 16GB model is the only model coming to Verizon. Sure, it has SD Card support, but as rushmore points out, if you're into games, that 16GB will go quick and you may want to opt for a phone with at least 32GB internal storage, whether it be an S3 or something different. (Media such as pics and video can go to SD Card, but apps cannot). If you're not really into game apps, than it really shouldn't be an issue.
Here's another response I gave to this subject.
There are probably other points to consider as well, but those are the primary ones I can think of right now...
So that's the bottom line. I personally think that the S3 will do just fine for quite a while, it's a great phone -- but if you're really into top of line tech, waiting for the S4 is definitely the better choice of the two.