GFCI Outlet Trips with Receptacle TEST Button, but not with GFCI Outlet Tester Tool
Grounded circuit (green/bare ground wire wired properly):


Push "Test" on the GFCI protective device, and it trips -> PASS 
Push "Test" on the GFCI protective device, and it does not trip -> FAIL
Push "Test" on a plug-in GFCI tester, and it trips -> PASS
Push "Test" on a plug-in GFCI tester, and it does not trip -> FAIL


A fail here indicates the GFCI unit is probably defective. 

Ungrounded circuit (green/bare absent or defective):


Push "Test" on the GFCI protective device, and it trips -> PASS 
Push "Test" on the GFCI protective device, and it does not trip -> FAIL
Push "Test" on a plug-in GFCI tester, and it trips -> FAIL FAIL FAIL
Push "Test" on a plug-in GFCI tester at the GFCI receptacle, and it does not trip -> PASS
Push "Test" on a plug-in GFCI tester at a downstream protected outlet, and it does not trip -> MEANINGLESS


Fail at the GFCI device probably indicates it is defective. 

Fail on the plug-in GFCI tester (i.e. it trips!) indicates they have bootlegged ground at the GFCI receptacle- attached the neutral wire to both neutral and ground.  It will seem to work at the GFCI, but is still dangerous.  

Bootlegging ground at a downstream GFCI receptacle is a mistake, because one of several electrical faults could put 120V on the the grounds, e.g. the cover plate screws or a machine chassis.  However this is difficult to detect, since a properly wired downstream receptacle will behave exactly the same way.  This means for ungrounded downstream receptacles, plug-in tester testing is completely meaningless. 

Once you have settled the question of bootlegged grounds, here's how you test an ungrounded GFCI.  Plug your GFCI tester into your handy dandy 2-3 prong "cheater" - the kind with a short green wire as a pigtail.  Extend that green wire all the way to a reliable ground source, e.g. the panel in the basement. Now, the GFCI tester should work normally, since you have rigged a proper ground to it.   

GFCI protection is pretty effective, and I would be confident in an ungrounded circuit if it has GFCI protection.  However if you are unable to get the external device to trip, you'll need to pop the cover off and see if the ground is present, missing or bootlegged.