I would like to allow another account to vote in governance on my behalf. What are the trade offs between setting up a governance proxy vs delegating voting power?
If an account sets a governance proxy, it basically allows the proxy to submit transactions related to any governance related pallet. These are Democracy, Council, Technical Committee, Phragmen Election, Treasury, Bounties, Tips, Utility and Child Bounties pallets.
On the other hand, delegating voting power only applies to the Democracy pallet. It consists of giving the voting and conviction power to another account for voting in referenda. So any democracy vote performed by the delegated account will result in an additional vote from the delegating account.
In summary, by setting a governance proxy you are giving a lot of more control of your tokens and action flexibility (dozens of different calls and functionality) than just delegating your voting power to another account (one functionality).
Finally, it is noteworthy that in OpenGov (Governance 2.0), the concept of multirole delegation is introduced. With this, you can specify a different delegate for every class of referendum in the system.