In your terminal's WiFi menu, you will see a checkmark next to one of your WiFi profiles.
However, the checkmark only indicates which WiFi profile is currently active. This checkmark will remain even when the terminal is taken out of range of the actual WiFi network. In order to have an internet connection, you should make sure that there is both a checkmark and a "WiFi" icon.
You can verify whether your terminal is correctly connected to a WiFi network by checking the 5 + Green info menu for an IP address, or trying the PSP Connection Diagnostic Test.