To paraphrase a wise man: "The M1, M14, and BM59 are very similar, except where they are different." So how different are they? That depends on what's considered "different." Here's some information for current or prospective owners.
The M1 rifle is a semi-automatic, gas-operated, clip-fed, shoulder weapon that weighs about 9.5 pounds. Whether it was built by Springfield Armory, Wincherster, Harrington and Richardson, International Harvester, or Beretta, nine times out of ten it will be found in caliber .30 (.30-06 or 7.62x63mm). In American service, some M1 rifles were converted to 7.62x51mm NATO for the U.S. Navy in the 1960's and are called the Mk 2 Mod 1. Original Navy conversions are quite rare and are collector's prizes. Unless so stamped with Navy nomenclature, 7.62mm NATO or .308 Winchester M1's are re-barreled rifles with spacer blocks added. [The spacer block acts as a feed ramp for the shorter NATO round and prevents .30 ammuntion from being loaded into the rifle.] Also, some Beretta M1's may have been converted, but I have not seen one.
The M14 rifle is a product-improved M1. It was designed as a select-fire, gas-operated, magazine fed, shoulder weapon that weighs about 8.5 pounds unloaded. Genuine M14 rifles were built by Springfield Armory, Winchester, H&R, and TRW (Thompson, Ramo, Woodridge) for the U.S. Department of Defense. As issued, most M14 rifles had all the automatic parts included except that the selector shaft was locked from functioning. The select-fire M14A1 [aka M14E2] was an attempt to make a controllable squad automatic rifle similar to the obsolete Browning Automatic Rifle, Model 1918A2. The M14A1 was not entirely successful in this role and was replaced by the M60 General Purpose Machine Gun (GPMG).
There have been several other civilian manufacturers of the M14 or M1A as it is called. Receivers may be found with names such as Springfield Armory (Geneseo, IL), Entreprise Arms, Armscorp, and M-K Enterprises (as well as some others) and cannot accept the selective-fire parts. Most of these are investment cast (except for M-K Enterprises that are forged). The civilian-designed receivers have dimensional differences from the original G.I. contract receivers. Norinco (China North Industries on the Chinese mainland) also imported M14-type rifles into the U.S. These guns are not heat treated to G.I. specifications and are too soft -- especially in the critical bolt lug and receiver areas. Some after market retailers, such as Smith Enterprises, offer a re-heat treating service for the Chinese M14S. Chinese rifle threaded parts will not interchange with U.S. parts because the Chinese threads are metric. Taiwan also built the M14 rifle as the Type 57 rifle. Both Norinco and Type 57 magazines feed and function like the original U.S. G.I. magazines. Norinco magazines are Parkerized but have no manufacturer's markings. Type 57 magazines from Taiwan are blued, unmarked, and are built on the same machinery that made the U.S. magazines. U.S. aftermarket magazines for the M14/M1A are mostly junk and neither feed nor function well.
The Beretta BM59 [later BM62] were modified M1 rifles made by Beretta. Beretta took the basic M1 design and converted it into a select-fire, gas-operated, magazine fed, shoulder weapon that was the tactical equivalent of the M14 and FN FAL. Beretta offered the customers of its license-built M1's the conversion service as well as building these rifles for the Italian military. What Beretta did was take the basic M1 rifle and modify it: one third of the parts remained original, one third were modified, and one third were new. This was an economical way to come up with a NATO battle rifle at far less cost than designing one from scratch or getting a license from FN or H&K or whoever. The BM59 ITAL was a paratroop version of the standard BM59 rifle that had a detachable compensator, a pistol grip, and a folding stock. The Beretta BM62 was a modified BM59 that did not have all the features of the former rifle. Notably the things deleted from the BM62 were Beretta's tri-compensator (a combination flash suppressor, compensator, grenade launcher, and bayonet lug) and the folding bipod. A shortened form of compensator was fitted that also acted as the gas cylinder lock for the gas cylinder.
Springfield Armory (now Sprinfield, Inc.) of Geneseo, Illinois, bought all the BM59 parts and accessories from Beretta went out of production on the rifle in the early 1980's. These Springfield-built BM59's are semi-automatic only, although they retain all of the original's features. Springfield also built the BM62 when the BM59 parts were in short supply. Parts kis for both rifles are sometimes available though a subsidiary of Springfeld called RSI, Inc. or Rock Island Armory.
The BM59 and BM62 series of rifles probably used the most expensive magazine ever designed for a combat arm. The magazines for the rifle were extremely strong and used a milled aluminum follower. When Springfield first started marketing its version of the BM59 and BM62 rifles, the cost of the magazine was $40 each! BM59 and BM62 magazines do not interchange with the M14 rifle, or any other rifle for that matter.