Hi. Sorry if I'm late to the party, but I hope this info will be useful anyway ...
BETA is the British Equestrian Testing Association. The WoofWear company is based in England, so they use the British standard. The BETA standard is the same as the European standard, EN 13158 (which you might see referenced on some body protectors). They have 3 levels of protection. Level 1 (the lowest) is for jockeys only. Level 3 (the highest) is recommended for most riding, including jumping, young/untrained horses, and riding on roads.
ASTM is the American Society for Testing and Materials. Companies that are in the USA are more likely to use the ASTM standard, which is ASTM F1937. There's only one level of protection in this standard. Some companies will have their body protectors tested to both standards, but that costs more, of course. Most countries except the USA and Canada use the BETA/EN 13158 standard. The ASTM and BETA/EN 13158 standards can't be directly compared to each other, because they measure different things. However, if you just look at the severity of the impacts that are used in the tests, then the ASTM F1937 tests are about the same severity as the BETA/EN 13158 Level 1 tests.
ASTM F1937: 24.5 Joules of kinetic energy. Personally, I like to wear a body protector that meets the BETA/EN 13158 Level 3 tests. The latest version of this standard was published in 2009, so you want the BETA:2009 Level 3. This will be clearly marked on the body protector. You want to check the label on the protector itself, because the information in the online catalogues is usually incomplete. The previous version was the BETA:2000, and the tests in this version are the same severity as the 2009 version. The newer standard has more requirements on covering the zippers and things like that. If your body protector says BETA:2000 Level 3, it's probably just as good, but is an older one. If you see a body protector that just says "BETA" or anything other than BETA:2000 or BETA:2009, then it was probably tested to an older version of BETA (Prior to the year 2000), and is out of date.
BETA/EN 13158 Level 1: 20 - 25 Joules of kinetic energy.
BETA/EN 13158 Level 3: 35 - 45 Joules of kinetic energy.)