R9-380 and GTX770 are in the same performance tier on tom's gpu hierarchy chart.
https://www.tomshardware.com/reviews/gpu-hierarchy,4388.html
Likely not particularly beneficial.
Do not select a GPU based on just specs.
AMD and Nvidia use vram differently.
VRAM has become a marketing issue.
My understanding is that vram is more of a performance issue than a functional issue.
A game needs to have most of the data in vram that it uses most of the time.
Somewhat like real ram.
If a game needs something not in vram, it needs to get it across the pcie boundary
hopefully from real ram and hopefully not from a hard drive.
It is not informative to know to what level the available vram is filled.
Possibly much of what is there is not needed.
What is not known is the rate of vram exchange.
Vram is managed by the Graphics card driver, and by the game. There may be differences in effectiveness between amd and nvidia cards.
And differences between games.
Here is an older performance test comparing 2gb with 4gb vram.
Similar video cards are often available in versions with more than one memory size. The GeForce GTX 680 is an example, and comes in both 2GB and 4GB variants. With computer components more is often better, but does doubling the memory on a video card like this actually help with game performance...
www.pugetsystems.com
Spoiler... not a significant difference.
A more current set of tests shows the same results:
http://www.techspot.com/review/1114-vram-comparison-test/page5.html
And... no game maker wants to limit their market by
requiring huge amounts of vram. The vram you see will be appropriate to the particular card.