There are two basic differences between AT and ATX power supplies: The connectors that provide power to the motherboard, and the soft switch. On older AT power supplies, the Power-on switch wire from the front of the computer is connected directly to the power supply.
On newer ATX power supplies, the power switch on the front of the computer goes to the motherboard over a connector labeled something like; PS ON, Power SW, SW Power, etc. This allows other hardware and/or software to turn the system on and off.
The motherboard controls the power supply through pin #14 of the 20 pin connector or #16 of the 24 pin connector on the motherboard. This pin carries 5V when the power supply is in standby. It can be grounded to turn the power supply on without having to turn on the rest of the components. This is useful for testing or to use the computer ATX power supply for other purposes.
AT stands for Advanced Technology when ATX means Advanced Technology eXtended.