Having two connection will always cause problem sadly. The problem is the VPN connect from one of the two ready and when the user request an site or share the computer got two default route, so it try by one of the two gateway available.. Another issue is when the VPN kick in, there is DNS available on that virtual card, but the second card got a DNS's server too (public's one). So when the computer use the bad DNS server it resolve \server as an unknown host, as it's not avaiable on the public DNS.
It's like having a house with two front door, you can't be sure the postman/ups man on which he will knock on.
I always suggest to the user to close the wifi to use a cable for latency issue, if no cable then make them use wifi. It fall more as users training.
Having two connection can allow you to make some high end scenario on the other side, but it's too complex for a normal users. (like default route for only compagny IP inside the VPN and internet traffic via the other cards, but it's hard to fine tune as it's specific per computer/user)