Different "Parent" cartridges -
The 243 Win was originally a .308 Win/7.62mm NATO (7.62X51 mm) brass case necked down to .243/.244.
The .244 Rem/6 mm Rem was originally a 7X57 mm Mauser brass case necked down to .243/.244
The difference is mostly in length, i.e. 51mm vs.57mm. OF course, the two different cases have different tapers to the body of the case.
So - NO, not interchangeable ammo, at all.
The only thing they have in common is the case head diameter, and the diameter of the neck (.243/.244).
Your dad's rifle, having its caliber stated as ".244 Rem" (stamped on the barrel at the factory) has a slow rate of twist in the rifling in the barrel, and will not stabilize the heavier bullet weights. The .243 Win has always had a faster rate of twist, which made it accurate with all bullet weights. Remington realized their mistake after a few years, and changed the rate of twist to the same as Winchester rifles, and re-named their rifles to "6 mm Remington".
Your dad's rifle, since it has a greater case capacity than a .243 Win, can be handloaded to about 100 fps higher muzzle velocity than a .243 Win, if you load the same bullet weight as the .243 Win.
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