Short answer: all three are excellent rifles, worthy of a good spot in your safe. I like the BM-62 best of all, as it is a lot handier without the 7" tricompensator on the end of the barrel (both are 17.4" bbls w/o muzzle device), and it is a semi-only and doesn't make it to the banned lists. I think it is one of the best handling carbine-sized rifles of it's type anywhere. If you like Garands, and you like the short M1A Bush/Scout rifles, you'll go nuts over this one. And it strips down much faster and cleaner than either. Why someone hasn't continued this exceptional design is beyond me. Down side: mags do not interchange with the M1A and cost $60 each.
The real Berettas are better finished than the M1As, and are built on forged receivers. But there are a lot of non-Beretta rifles out there, so it is a buyer beware situation.
The long answer version: The real real Beretta BM-59 would be a select fire Class 3 firearm and and there were very few available for sales to civilians. That tricompensator is highly effective, and it is way more controllable in full auto than the M14 (see http://myweb.cableone.net/uziforme/Beretta.WMV
), but the extra 7" is less desirable in a semi. Some kits for modifying Garands did make it to the civilian market, so converted Garands exist in a number of BM-59 variations. Do NOT believe something is a BM-62 of BM-59 just because it is advertised as one, and do not pay Beretta prices for non-Berettas, although the Reece/Springfield version looks decent, and uses most good parts. Beware of buildups using Garand gas cylinders from other mfgrs -- some of them will have the problems "Tankers" have. The genuine Italian-made Berettas were imported by Berben Corp, NY, NY and are marked "Made In Italy" -- these are the real thing made with typical Beretta quality.
The BM-62 is a clean, semi-auto-only version manufactured for the civilian market. Even in preban days, lacked the extended "tricompensator" (*much* longer, combining a flash hider, compensator designed to control full-auto firing, and grenade launcher), grenade launcher sight, all the select fire parts and receiver differences (if you have a real full-auto BM-59), as well as the bayonet lug, bipod, and winter trigger. [Since it was designed without these "evil features", and didn't have a pistol grip, it didn't make the banned lists.] The muzzle has a short, solid (uncut) muzzle brake-like attachment which had no holes cut and no bayonet lug, and the stock is a civilian version without the selector, connector, and folding bipod cuts.
Being shorter and lighter, the 62 handles better than the 59, but is fully functional, and lacks only a good muzzle break, which you can get a 'smith to cut for you (try Smith Ent.).
The stock had no cuts for the full-auto mechanicals or bipod, and the whole thing looks a lot "cleaner" since it lacks the "tricompensator" and the unnecessary stuff SA hung on their versions. Since they are marked BM-62, not BM-59, and are built specifically as sporting arms, they have fewer regulatory issues connected with having a gelded version of a full-auto rifle. The BM-62 was the favorite of the late Mel Tappan. You can see pictures of BM-59, 62, and 69 models (and buy 'em) at http://www.reesesurplus.com
-- note the gap between the gas cylinder and the barrel. Reece also sells BM-59 parts and magazines, and quite a few trigger group parts interchange with the M1Garand and/or M14 / M1A.. Reece is the family which owns Springfield Armory, Inc. and produces the M1A. There are also several variations with different barrel lengths, E2 or folding stocks, and different compensators. There is also a BM-69 hybrid which is basically a BM-62 with a bipod and compensator.
You can find highly detailed information in the Beretta Technical Brochure (4-language PDF) -- the index is at http://www.berettaworld.com/index.aspx?m=53&did=38
Also download this PDA file: Beretta's BM59: The Ultimate Garand, 2002 GUNS Magazine Combat Annual available at: http://gunsmagazine.com/bm59.html
Hope this helps