The VPN application does not provide its own private browser. When the VPN is enabled and running it is providing your computer with system-wide protection which means all internet traffic on your system is being protected. This means that you are able to use any browser that is downloaded on your system.
If you are looking to achieve the closest browsing experience that you had with the USB key, it may be best to use Mozilla Firefox when SurfEasy is enabled. The USB key used an older browser version of Mozilla Firefox so you may be more familiar with the settings and functions of this browser. Firefox also allows you to set your browser to never remember history, as it did in the private browser. Not all browsers provide this option and if you are using a different browser that does not provide this option it may be best to browse in private browsing mode or incognito mode to ensure that no history or other browsing data is saved.
You are able to open a private browsing window in your browser by pressing Ctrl + Shift + N after launching your browser.