Yes, you can have both apps open at the same time. They are accessing the same information in the IDE for your one Samsung account. In the NEW app, your Hub actually shows up as a Device in the Devices List. If you were to rename the name of a SmartApp in the Classic App, the next time you view Automations in the NEW app, that SmartApp will reflect the name change.
The above statement applies if you are logging into both the SmartThings Classic and SmartThings (Samsung Connect) applications with the same Samsung account.
If you are logging into SmartThings Classic with a SmartThings account and logging into SmartThings (Samsung Connect) with a Samsung Account (only way to get into new app), that’s an entirely different scenario. That means you haven’t yet migrated your ST account to a new/existing Samsung account which would reflect two different places in the IDE.