Does GPU with more memory provide better mining for Ethereum? Here is a Quora an answer:
In general, 4GB is sufficient for even the most demanding hashing algorithms and you will not see any performance gains just by having more memory. This is where the speed and quality of the memory comes into play. The 8GB RX 480 has a higher memory clock speed which gives it a slight edge while running memory intensive algorithms such as Ethash (used by Ethereum). When running at the same core clock, the hashing speed of a 8GB RX 480 can be higher by 25–30 hash/sec compared to the 4GB RX 480.
However, hashing speed also depends on the quality of the GDDR5 memory. Samsung based chips seem to perform little better (20–25 hash/sec) compared to Hynix or Elpida. They are also easier to overclock.
What about Radeon RX 580, which one is better? In what cases should I consider memory?