What are the monetary or reward benefits of running this node long-term?

From the node itself and programmatically, nothing per say directly if not configured for that specific purpose (although it can, but I would not advise into this type of infrastructure).

An API Infrastructure "includes" (but I would run it separately for security reasons) what's called the HiveD code (basically the Hive Blockchain core code) that can be used to verify blocks (witnessing). When you enable your node as a witness node, people can then "vote" for your witness and once you node is scheduled to verify a block, it will do it, and gain some HIVE reward in form of Powered Up HIVE.

Having that said, and this is my opinion, when witnesses also run Hive API nodes, they are helping with decentralization for the entire ecosystem, as the more decentralized infrastructure exists across the world, and that should help with the apps independence which can take advantage of more nodes for their calls (NOTE that big/complex apps should ideally run their own Hive API nodes, although that would not be technically required).

And "naturally" if a witness is helping the ecosystem, that will potentially suggest the Hive community that you are trying to help the blockchain, and that might result on people to potentially vote for that witness and the higher you are on the witnesses list (excluding the top20 where everyone produces the same amount), in what is called, "a backup witness" the more regularly you will sign blocks, and therefore more HIVE generated from your node block verifications.

Does running a public API node itself generate any direct income?

Hopefully I answered this above.

Are there infrastructure incentives from Hive, or is it mainly reputation + potential future votes?

You can create a proposal if you think your support to the community should attract payments from the DHF (a DAO voted by the community Staked HIVE). You can check current proposals here https://peakd.com/me/proposals (anyone can create proposals). Once a proposal is above what's called the "Return Proposal", you can start receiving the funds requested from the DHF pool.

Reputation is (in my view) is one of the hardest things to protect as a witness and also something very easy to get damaged if you are not responsible for your actions and don't do your best to protect the chain.

In conclusion, being a witness, and running an API infrastructure does not per si get you more votes or rewards. But depending on what you do to help the ecosystem, people will accordingly curate that value towards their interpretation of what you do.

Although this is probably one of the biggest privileges I ever had on participating on a decentralized worldwide blockchain where I have leaned tons on!

Hope this helps. And if you are interested on running a node "just to give it a try" let me know.