As a peer:

• You can derive the approximate location (GeoIP for IPv4, latencies and allocations for IPv6) unless the node is operating as a hidden service.
• You can query the node for it's version number.
• You can query the node for current date and time.
• You can query some of the nodes recent connections with the addr host discovery mechanism, giving you the IP address and the time the node last saw the peer.

As someone with universal access to a nodes connection:

• You can see which transactions (and addresses) belong to a user.

That's about it really unless you want to start poking around for other services, such as a web server on port 80 which would give you a lot of information about the owner, but only if it exists. bitcoind is designed to leak as little information as possible.