I know I'm resurrecting an old thread a little, but since it's directly related to what I've been dealing with I didn't want to make another thread for the same...
So dealing with pretty much the same (at current test build of 10.8.3) I found out the following.
When you move your VPN connection to the top of the list (Set Service Order) so it will be above your regular connection, or when you select "Send all traffic over VPN connection" in the Advanced options - youll get everything routed through the VPN connection which will in most cases break your internet connectivity (when VPN is connected) and even if not, you'll have an unnecessary overhead in the network path... That's what one would expect so nothing strange here.
However what to do when you want to have both at the same time...
Because of the previous limitations I had to keep the VPN connection down in the list. That let me to use the internet while at VPN, however there was a little problem with the VPN connection - access (to our company machines) by IP address worked for me, however access by hostname (and thus DNS name resolution) didn't.
It seems that in such cases the DNS cache needs to be reset (causing the new DNS resolving order to be enforced) otherwise my client applicatons (SSH, MS Remote Desktop Client) don't know about the new DNS servers that become available via the VPN connection (they appear in the DNS configuration listing but nothing wants to use them).
Once you reset the DNS cache everything starts working properly.
So basically it seems that VPN client in Mountain Lion should do this automatically at the end of establishing the VPN connection, but it doesn't. I consider this to be bug and will try to report it.
There could be some workarounds such as adding the company DNS servers to your default list (/etc/resolv.conf or in the network options) however I didn't try that as I decided to live with IP addresses for now because I don't consider any of these workarounds as very "clean" and it isn't that big trouble for me (for now).
Some resources:
Viewing the current DNS configuration (basically the resolving order): scutil --dns
https://developer.apple.com/library/mac/#documentation/Darwin/Reference/Manpages /man8/scutil.8.html
Resetting the DNS cache: sudo killall -HUP mDNSResponder
http://support.apple.com/kb/HT5343
By the way I'm pretty sure this worked fine in Snow Leopard. The only thing I did differently was that I had my VPN connection at the top of the connections list. Then both VPN and local network + internet worked fine for me.