"Compact" aluminium or copper conductors have a smaller external diameter than conventional wires, this is achieved by shaped wire strands that give less wasted space between the strands.
If code requires that #4 be used, then at least #4 must be used, BUT compact #4 will have a reduced external diameter and may permit smaller conduits.
In practice I have my doubts as to the usefulness of such conductors. They tend to be less flexible than conventional wire and therefore potentialy problematic to pull into conduits.
Code may permit of smaller conduit, but that does not mean that it is a good idea in practice.
Also I would expect that these wires would run slightly hotter if heavily loaded.
A "compact" #4 and a conventional #4 would have the same resistance and therefore the same losses in watts per foot.
However the reduced diameter, and therefore reduced surface area of the compact wire means that it must get slightly hotter with same watts per foot losses.