If you do any banking or other financial transactions online, you may find that disallowing Flash cookies prevents logging in to some sites. These would be sites which use a personalized image-and-caption combination to guard against phishing attacks: when your computer is recognized, the combination is displayed, and only then do you enter your password.
I access Vanguard (with Firefox) and Ally Bank (with Safari), both of which use the login sceme you've described, and neither has ever tried to stick me with a Flash cookie.
Edit: Note that the option in the Flash pref pane is to block Local Storage, and that differs from Flash cookies.
Edit 2: In fact, I've got a Flash cookie for each of the three sites for which I've specifically blocked Local Storage.
And, further, the Flash pref pane allows you to set your prefs
by site in both its "Local Storage Settings by Site" and "Camera and Microphone Seeting by Site" sub-panes.
Edit 3: And, finally (I think), Adobe has given us the option to block assorted stuff, but Flash cookies are
not included in the mix.