Well, neither device has had its specs completely confirmed, so this is based entirely on speculation, though it should be fairly accurate.
First and foremost is the version of Android. The Vigor will have some variant of 2.3 while the Prime will have 4.0 Ice Cream Sandwich. ICS should be bringing a lot of big changes (and hopefully improvements) to the Android platform.
Both are expected to have a 720p (1280x720) display, the Vigor at 4.3 inches, and the Prime at 4.65 inches. This may suggest that the Prime is bigger overall, but because ICS will do away with the capacitive buttons, the two phones will be very close to the same size. I imagine the Prime will actually be thinner.
The Vigor will probably have a Super LCD screen, which gives immaculate viewing angles and excellent color accuracy. The Prime will be Super AMOLED Plus, which gives very vivid color and is generally easier on the battery.
As far as processor is concerned, it's a bit of a toss up because no one really knows which processors either phone will be getting.
Odds are that the Vigor will have the Snapdragon MSM8660, a dual-core CPU clocked at 1.5GHz paired with the Adreno 220. However, it's possible that it will have the Snapdragon MSM8960 which is also a dual-core 1.5GHz processor but is better optimized for battery life and has the Adreno 225.
The Prime will probably be getting the OMAP 4460 which is a 1.5GHz dual-core processor with an SGX540 GPU. The SGX540 is a bit concerning since it is showing some age, but in the 4460, it will be clocked at 384Mhz instead of the usual ~300MHz.
The MSM8660 (with Adreno 220) and OMAP 4460 (with 384MHz SGX540) should perform fairly similarly in games and other graphically intensive tasks. The MSM8960 (with Adreno 225) would easily trounce the SGX540, though.
There is also rumor that the Prime will have the a 1.5GHz dual-core Exynos processor. This would have the Mali-400 GPU. On paper, the Mali-400 bests both the Adreno 220 and 225; however, graphics drivers appear to be a bit immature right now, so real world performance is a little sub par (though still very good).
What I would love to see is the OMAP 4470 in the Prime. It, too, is a 1.5GHz dual-core processor but it has the SGX544 GPU. This would beat out the Mali-400 in the Exynos by a fair margin. But this is by far the least likely processor we'll see in the Prime.
Those are the major points. Hope it helps.