The adjustable die is used to make different thread tolerances. Thread tolerances are expressed in "H" limits. A thread made to the exact size of the nominal size is an H0 (as in zero). Each .0005" over the nominal size is one H size larger. For an example a 3/8-16 H1 thread is .0005 larger than the .375" nominal size, an H2 thread is .001" larger, and an H3 thread is .0015 larger" If I recall correctly H limits go up to H12. Those over H4 or H5 are usually used when threads are cut then the part is plated.
The adjusting screw allows the die to be enlarged to meet the H limit required. The most common for fractional sizes is H3. Smaller number sizes are usually H1 or H2.
As for hex dies being for rethreading and round dies being for thread cutting that is a general rule, but not necessarily a hard and fast rule. I have hex dies that are thread cutting dies. They're generally used in tight places with a socket and extension where a standard die wrench would be difficult or impossible to use.
I have a full set of Snap On "triple hex dies" like this:
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They are thread cutting dies that can be used in either a die wrench or socket.