Edit: I feel bad for saying IPv4 sucks. It's one of my favourite pieces of tech and an astonishingly good one at that. It just doesn't have a big enough address space.
If the plane network uses 10.0.0.0/8; and then the VPN you're trying to connect to uses 10.0.0.0/8, stuff breaks.
Now, I think VPN applications are smarter than that and will still get the outgoing packet to the default gateway (citation and research needed), but what happens when it doesn't know to handle a route automagically. For instance, with DHCP, a router can tell your computer what DNS server to reach out to. If that's on the local network, now you see all DNS requests actually routing into the network on the other side where you almost certainly aren't going to be talking to a DNS server. And now, you can't go to any websites.
Hopefully this helps. I'm not the most knowledgeable about VPNs and routing, but I'm pretty sure this is all fairly accurate.