Automatics the shifting is precise, preset and quicker than a manual transmission. The advantage for manual transmissions is that the driver has more control when the car shifts so the driver can use more of his powerband, and before he tops out or before the powerband can drop out, he can shift into the next gear's powerband. Basically he can shift into the powerband rather than before it. Does that make sense? I know there is a better explanation for it.
The 03/04 Cobra models were equipped with the T-56 6 speed transmission. All other models were equipped with one form or another of the 5 speed. The 5 speed models included the T-5/T45. The "T-" transmissions were Borg-Warner Transmissions. Then there was there was the Tremec 3650 which was found in the 01-04 GT, Bullitt, and MACH models. Tremec took over Borg-Warner in 1998, so (at least for the New Edge body style) BW and Tremec are the same company/made by the same company.
I don't know if the Tremec 3550 was in any Mustang or it's successor the TKO 500/600.
As for the 5 speed's strength I do not know how it compares off the top of my head, but Tremec says the 3650 has a Torque Capacity of 360 ft/lbs (390 for one particular model).
The T-5 was rated for anywhere from 175-300 ft/lbs Torque. It is noteworthy that there are Foxbodies which are equipped with the (more likely World-Class or WC) Borg Warner T-5 which are well above the 300 torque mark. So they can be quite strong.
The T-45 was used in the '96-'99 Cobra and was rated for 325 ft/lbs Torque. It was only 5 gears.
Also in mid 2001 production the Ford replaced the Mustang's Transmission from the T-5 to the 3650.
Hmmm... I think I will be attempting to start on a Transmission article now
As for the Automatic Transmissions, 1994-1995 saw the AODE transmission while the 1996-2004 had the 4R70W, but I don't have the specific details in front of me and will leave that up for another discussion.