I have had a lot of questions regarding the differences between two of Nike's new awesome shoes. Here are my thoughts.
1. The uppers are different. The Run+ 3 has a tongue and while being a little heavier in the upper it is probably slightly more breathable with more mesh and less synthetic material. The 3.0 V4 will give you more of a sock like feel and is slightly softer inside. The Run+ 3 has the lace lockdown system while the 3.0 V4 does not. I haven't found this to be especially important though.
2. The sole is the key difference between the two shoes. The original line of thinking for the Frees was that barefoot running is a 0 and running in a traditional trainer is a 10. The Free Run+ 3 is a relative of the original Free 5.0 and the Free 3.0 is slightly more towards a barefoot shoe. It has an overall lower profile and you can feel the ground more. This new 3.0 is probably the most popular shoe in the minimalist category. It is far from a barefoot shoe though.
2a. Also, there are a few more differences on the soles. Both of these shoes are the first in the Free line to incorporate diagonal grooves in the arch area of the foot instead of the typical grid like pattern. These are another factor that push the 3.0 V4 more into the minimalist category The 3.0 V4 does have more of those grooves as well as a larger groove that runs most of the length of the shoe.
3. Heel drop: 4mm for the 3.0 V4 and 6mm for the Run+ 3.
4. Nike+: The Free Run+ 3 is Nike+ compatible but the 3.0 V4 is not. This is due to the lower profile in the 3.0.
What do I recommend? Both! I really enjoy both of these shoes. Of course, the Run+ 3 is closer to a typical trainer so if you are just starting the minimal route it might be wise to get one of these. If you have already tried out the 5.0/Run+ line and want more minimalist then maybe the 3.0 V4 is your best bet. Like anything else, I urge moderation and patience when beginning a minimalist regiment. It is stress on your body that it is not used to.