I have savings accounts with both of them. The first time I made a large withdrawl (over 5k) from American Express I received a phone call asking if I had authorized the withdrawl. Although I generally don't make frequent withdrawls from savings, I've never experienced any issues with Amex needing to call me for verification after that first time. Between Amex and GS, Amex has better customer service.
You can go here https://www.depositaccounts.com/savings/ and do some research and decide for yourself
My advice is a hybrid approach:
(1) EASE OF USE: Choose a bank that has a reasonably high interest rate (anything over 2.0% qualifies at the moment), but making money from interest is not your chief focus. Instead you are looking at the other features you mentioned: getting money in and out easily, customer service, web site design, and whatever else is important to you. You will have a wide selection to choose from: the two you mentioned but also Discover and many others.
(2) REVENUE: Open a few bank accounts that have big signup bonuses each year. Interest rates have certainly gone up in the last 12 months, but bank bonuses still outstrip what most people could earn. Opening a few bank accounts a year can get you $1000 easily ($2000 if you work at it) whereas $10,000 at a 3% interest rate will get you only $300.
Looking for a no fee high yield savings account that I can move money in and out of easily without $1000 max limits or something like that to get the higher %. GMS is at 2.25% vs AMEX 2.10% currently, but opinions on who you would go with?
I have had experience with four high yield savings.
Looking at the rates over 9 months averaging both savings and 12-month CD's, they each have high and low periods but over the long run
GS and Barclay's are on top, next Ally and C1 last.
Have not tracked AMEX
Capital One Money Market: Min $10,000, Slow Transfers, OK-Web, OK-Customer Service, Checking, So-So Interest
Ally : Checking, Good CS, Slow Transfers, OK rates
GS : No checking, Poor SC, Poor Web, Slow transfers, Top Rates
Barclay's : No Checking, Good Web, Good SC, Fast Transfers, Top Rates
Because of transfer speed, web interface, features, customer service and overall interest earnings,
I only have Barclay's at this time.