One of my vegetable garden areas has low organic matter. I would like to help it with a grass winter cover. The two I am looking at are oats and rye, both of which grow all winter and get plowed under in spring. My question is, which one is better for poor soil? I know they both grow well in my area.
I think the answer depends on more factors than what you've provided in the question -- especially your specific goals, but I can lay out the considerations I would make in choosing between the two.
According to a UVM factsheet, rye can add 5 tons of organic matter per acre. I like rye because the seed is cheap, it germinates quickly and reliably, you can start it late, it catches reliably, it always survives the winter, and it goes like gangbusters when the snow melts in spring. The only real downsides I can think of are (a) it is so vigorous it can be challenging to till under in the spring, and (b) it is allelopathic so you can't sow seeds for a couple of weeks after tilling.
I have less experience with oats -- and none as a winter cover crop, only spring seeded. According to a SARE document, oats can add 1-2 tons per acre with fall seeding. Locally for me, oat seed is less available than rye. Also for me, oats will reliably winter kill, which means it would need to be started earlier in the fall -- but it also would be easier to incorporate in spring. Oats also have allelopathic properties; I think winterkilled oats won't cause problems in the spring. Because of the allelopathy, you should wait 3 weeks (according to SARE) before planting after tilling the oats under.
Either cover crop makes a good nitrogen trap -- "soaking up" excess nitrogen left in the soil following the previous main crop and storing it for the following season. They both do a good job of smothering weeds.
The SARE document on oats linked above mentions in "Comparative Notes" at the bottom:
[Compared to Oats,] Rye grows more in fall and early spring, absorbs more N and matures faster, but is harder to establish, to kill and to till than oats.
So I guess if you've got a good rototiller and are willing to work at tilling it under in the spring, rye would be my recommendation.