Just scanned the prior replies, so may have missed what I'm going to write. No difference between the 244 or 6mm remington cartridge. Your 244 Remington rifle has a slower twist rate in the rifleing than a Remington rifle stamped 6mm Remington. What this means for you and your ammo selection, is that you need to shoot lower grain (weight) bullets in your .244 for accuracy reasons. The .244 Remington was designed for shooting varmint sized critters, therefore smaller/lighter bullets used. Folks that reloaded, tried using heavier bullets to use cartridge for deer hunting, for example. Unfortunately, the heavier bullets proved to be fairly inaccurate in a .244 Remington. This problem caused the .244 to be an unpopular cartridge and its main competitor was the .243 Win which could shoot more accurately the heavier bullets due to it having a faster twist rate. Remington to correct its problem, will rename the 244 to 6mm Remington and Remingtons rifles will be made with a faster twist rate and barrels of these rifles stamped with 6mm Rem. Basically, its thought that Remington messed up by marketing its 244 rifles as only varmit rifles and making them with a slow twist rate. Actually, the .244/6mm remiington is a fine cartridge that has a slight edge ballistically on the .243 Win. However, today, no one makes a rifle for a 6mm Remington, but darn near all rifle maker sell a .243 Win.
Here is your problem. Only ammo today that you can probably find is 6mm Remington and it has the heavier bullets. You may not like the accuracy of your wonderful 722 Rifle with them.