The Following are Quotes From Wikipedia regarding the two rounds,
30-06
"The .30-06 Springfield cartridge (pronounced "thirty aught six") is a .308 inch (7.82 mm)(.300 inch is 7.62mm) caliber rifle round, also known as the 7.62x63mm, introduced to the United States Army in 1906 (hence "-06") and standardized, with use continuing into the 1960s, tapering off in the 1960s and early 1970s.
It was developed from and superseded the nearly identical .30-03, having a slightly shorter casing and a higher velocity spitzer bullet. There were three main production runs of the round, that roughly correspond to large stocks built up during wars: the initial .30-06, the M1 Ball, and the M2 Ball.
The M1903 Springfield rifle, introduced alongside the earlier cartridge, was quickly modified to accept the .30-06. It replaced the 6 mm Lee Navy as well as remaining older calibers such as the .30 US Army (also called .30-40 Krag) used in the Model 1892 Krag. The .30-06 remained the US Army's main cartridge for nearly 50 years before it was finally replaced by the 7.62 x 51 mm (7.62mm NATO, commercial .308 Winchester) with the adoption of the M14 in 1954. However, the first M14s were not fielded until 1957, and the .30-06 remained in service into the 1970s, mainly as a machine gun cartridge.
It was used in the bolt-action M1903 Springfield and M1917 Enfield, the semi-automatic M1 Garand, the Browning Automatic Rifle (BAR), and numerous machine guns. It served the United States in both World Wars and in the Korean War, its last major use being in Vietnam. Large volumes of surplus brass made it the basis for dozens of commercial and wildcat cartridges, as well as being extensively used for reloading. The .30-06's power, combined with the ready availability of surplus firearms chambered for it, and so demand for commercial ammunition, has made it a popular hunting round. It is suitable for large mammals such as deer, elk, and moose."
.308
"NATO's 7.62x51mm rifle cartridge, commercially known as .308 Winchester, was introduced in the 1950s as a standard firearm cartridge among NATO countries, though it has also become popular among civilians. The round is produced by many manufacturers in types that include tracers and blanks.
The military 7.62x51mm cartridge is nearly identical to the commercial .308 Winchester. NATO controls specifications for the military round while SAAMI controls specifications for the civilian round. The organizations have established two differences: the standard pressure is 50,000 psi for most military rounds, while the SAAMI maximum is 62,000 psi for the civilian round. The NATO M60 High Pressure Test round is loaded to a pressure of 67,500 psi, so military arms should be capable of handling the pressure of civilian rounds[1]. The military chamber is specified to be 1.645 inches, compared to the civilian chamber of 1.632 inches. Though this difference is just 0.013 inches,[2] a chamber of acceptable military length will put excessive stress on the thinner civilian brass, causing premature head separation. While it is not unsafe to mix .308 and 7.62x51mm ammunition, fewer problems will be encountered if the correct caliber designation is used.
The cartridge was introduced to military service in rifles and machine guns. It was used in the M14 Rifle and M60 Machine gun in U.S. service in the late 1950s. Fabrique Nationale's FAL became the most popular 7.62 NATO rifle in Europe and served into the early 1980s. The M14 was superseded in U.S. service as the infantry adopted a new round with the M16. However, the M14 and many other firearms that use the round remain in service. The round is used by infantry and from ground vehicles, aircraft and ships. It is used in the GE M134 Minigun as well."