Rice will never be able to appreciate space and passing between the lines like Carrick could.
Rice has far better physical attributes, and is probably better defensively given he started as a CB by trade. For a team that wants to predominantly play on the front foot though, you'd want Carrick. There are so few midfielders like that, which is why they're always lumped in together, often unfairly (Xavi, Scholes, Xabi Alonso, etc.) Carrick wasn't the best of that school of midfielder, but he's as close as England have come to it since Scholes himself. We were so lucky to have both of them together.
Put it this way, if Pep needed a new first XI deep midfielder and he had a choice between Rice and Carrick, I think he would almost always choose Carrick.