It's not about memory bandwidth, but throughput capacity and encoding capacity of UVD engine on these cards.
R9 390 is a higher revision of GCN 2 (or 1.2) with UVD 4.0, while
R9 380 is a GCN 3 (or 1.3) with UVD 5.0. Fiji, aka Fury, is UVD 6.0.
UVD is a special part of AMD GPU or CPU that is used to hardware accelerate video decoding or encoding. This affects what codecs it can record with and at what resolutions/frames per second.
For example, my old r9 270 couldn't do hevc and 4k, but my current rx 480 can, even tough they both have 256 bit gddr5 memory.
links to specific sections of aforementioned article, r9 390:
while r9 380 is weaker than r9 390 in raw power and sheer amount of compute shaders, it's still slightly newer as a chip.