I went to both within the last week.
Bread - Shamshiry used to bring their flat bread out cold in a ziplock bag. It's now served in a basket but it's the same cold flat bread. Amoo's serves their bread soft and warm - probably fresh baked too.
Green Sauce - both are excellent, with Shamshiry's being a little spicier and more garlicky.
Cornish Hen - both are tender, with Shamshiry's slightly more tender. And I think Amoo's sauce is a little more sour/acidic.
Rice - both are excellent, with slight edge to Shamshiry.
I also tried some Chilean seabass kebab and hummus at Amoo's. The fish was perfectly cooked, very tender, but it had the same seasoning as the cornish hen (which was on the sour/acidic side). The hummus was nice and smooth, and tasted like there might've been some olives pureed in there as well.
At Shamshiry, I also had a kubideh, which was excellent. There was a time that I avoided Shamshiry but quality is very good now. At Amoo's, it family run and the service is fantastic.