In-browser Kali Linux machines are exactly the same as any other Kali virtual machine with one exception: data persistence. When a new session is started, a machine with a fresh install of Kali Linux will be provided. This means that after a session is over, all additional configuration changes will be reverted and any data saved on the machine will be lost.
The VPN will not give you access to the same exact machines as your in-browser. You will be able to simulate the same attacks but you will not be working in the same environment. Moving between VPN and in-browser will result in reverted machines and brand new environments
Because in-browser access to your Kali Linux Machine is an alternative to the VPN connection, it resides in a different environment. This means that switching from VPN to in-browser Kali will result in a new environment (e.g., fresh target machines and Proving Grounds environment).
If I switch from VPN to in-browser access, will I see the progress I made while I was connected to the VPN?
Yes, you can see progress you made on the machines while you were connected to the VPN in the machine list. In the machine list, progress will be shown for each machine, as it would in the control panel.