Yes, the M14 rifle is a product-improved version of the venerable M1 Garand.
During WW2, reports came back from Army Ground Forces to Ordnance that the troops would like to see improvements in their M1 Garand rifles. One thing that they wanted was to replace the 8-shot clip with a detachable magazine. Another thing they wanted was lighter weight and selective fire. In response, Springfied Armory began the T20 rifle program.
The T20 series rifles started with the basic M1 and added a selective fire mechanism (almost identical to the current M14), used the M1918 Browning Automatic Rifle 20-round magazine, and had some other differences. Had the war gone on past 1945, Springfield had plans to put T20 rifles into production and into the hands of troops in 1946.
The end of the war, put the T20 rifles back into development and led to the T44 series of rifles. [John C. Garand's last project before he retired from Springfield Armory was the design of the M14's 20-round magazine. To this day, the M14 magazine has the lowest stripping force of any similar capacity magazine in 7.62 NATO caliber.] The T44E4 was type classified Standard A by the Secretary of the Army in 1957, beating out the conteneder, the Belgian FN FAL (T48).
Springfield got the T44 adopted as the M14 by telling the Army that it could reuse old M1 tooling as a money saving benefit. In reality, the old M1 tooling was too worn and would not hold the tolerances of the new rifle. New tooling specific to the M14 had to be made and this led to production delays getting the M14 into service.
There were three pther major contractors for the M14 rifle, besides Springfield Armory: Winchester Repeating Arms, Harrington and Richardson, and Thompson-Ramo-Wooldridge (TRW). Of these rifles, the TRW guns have the highest reputation for quality. Both Winchester and H&R had production problems that Springfield labored mightily to solve. TRW, on the other hand, set up a new factory with state-of-the-art computer-controlled machines to build the M14. Not only were the M14's built faster, they were built better than the M14's made by its competitors that still used single-purpose, manual machines.
Nevertheless, there were never enough M14 rifles to equip the American armed forces in the early 1960's and the older M1 rifles continued to soldier on. The war in Vietnam and other factors such as Project SALVO -- that looked at how soldiers actually fight in combat -- led to a different rifle from the M14. Project AGILE sent limited numbers of Armalite AR-15 rifles to Vietnam for evaluation in combat. Results were glowing and SecDef McNamara ordered a "one-time buy" of AR-15's (now called M16's) for the U.S. military. At the time, SecDef McNamara considered both the M14 and the new M16 as interim rifles because the Army had a new "wonder weapon" in-waiting that would replace both. This weapon, the SPIW -- Special Purpose Individual Weapon -- was a real Buck Rogers gun.
The SPIW was made of plastics and aluminum with some steel parts. It was very lightweight in the first rifles and fired 13-grain darts called "flechettes" that looked like nails with fins. The SPIW could fire both single shot and bursts of three, as well as full automatic. The idea of the three round burst was to enhance the soldier's chance of hitting targets in combat conditions. Later versions of the SPIW added a grenade launcher to the rifle. Some launchers were in 40mm and others in 30mm. One of the 40mm single-shot versions was adopted for use with the M16 rifle as the M203.
In the end, the SPIW program was canceled. McNamara canceled procurement of the M14 and closed Springfield Armory in 1968. The M16 replaced the M14 as America's standard military rifle and has been continuously developed and improved ever since.
The M14 rifle remains in-service with Ameican forces to this day as a special purpose rifle whose heavy caliber -- 7.62x51 NATO -- is useful in removing difficult targets at longer ranges than is possible with the smaller caliber 5.56x45 NATO M16/M4 rifles. The M14 is a tribute to the old way of building rifles: strong steels, forgings, and wood before plastics and lightweight alloys became standard.