First of all, let me address an important fact:
The core of the system has stayed the same. They're all point-buy character creation systems devoid of classes and levels. They're all skill-based. They all feature 3d6 roll-under task resolution. They all have an extensive collections of Advantages, Disadvantages, and Skills. They all have combat that scales from basic and abstract to advanced and tactical. They all feature 4 character attributes:
- Strength (ST)
- Dexterity (DX)
- Health (HT)
- Intelligence (IQ)
All are built to handle any genre, any character, and any situation you can come up with.
I'm going to limit my answers to the differences between 3rd and 4th editions, because I assume those are the ones you're really asking about.
3rd Edition had hit points were based on Health, fatigue based on Strength.
4th Edition reverses that.
3rd Edition attributes were all purchased on the same graduated scale of point costs that made mid-range scores cheap, and high-range scores expensive.
4th Edition attributes are purchased on two separate scales (one for, IIRC, DX and IQ; the other for ST and HT) with constant steps between scores.
3rd Edition was a very solid edition, but had certain problems. Probably the most talked about were IQ-related. For example:
- It was impossible to have an intelligent character that spoke a foreign language badly.
- Intelligent characters tended to be able to avoid the effects of mental disadvantages with ease.
4th Edition addresses those and similar issues. Usually, the fixes are integrations of rules that saw years of use and testing under 3rd edition.
3rd Edition also had a huge library of support books. But that meant that rules were scattered all over those books. At the very least, you needed the Basic Set and the two Compendia.
4th Edition has all the base rules in just the two Basic Set books - Characters and Campaigns. The support library for 4th Edition is smaller, but characteristically excellent. And the vast majority of 3rd edition material can be used with 4th.
3rd Edition had a solid implementation of variable damage types - Crushing, Cutting, and Impaling.
4th Edition damage has been expanded to include even more types, whose effects and even names escape me at the moment. But suffice it to say, in GURPS you'll never have that problem where a .45 and a battleaxe have approximately the same effect.
If you're looking for a recommendation, that's easy. Go with 4th. It's a finely-honed realization of the promise of GURPS - Any genre, any adventure. You can use most 3rd Edition books (many of which are available as PDFs from e23), and the 4th Edition library covers most of the major bases at this point - Fantasy, Space, Supers, Time / Dimensional Travel, Mysteries, etc.