There are several notable differences between the RPi A, and A+. According to the official release, the differences are as follows:
An expanded GPIO header. The A+ has 40 pins which considerably larger than the original 26. Additionally, the new form factor and I2C compatibility allow standard Raspberry Pi hats to be used with the A+ model.
MicroSD. The A+ switched from a standard SD card to a MicroSD card as its primary storage medium.
Less Power. The A+ has significantly lower power consumption due to more efficient circuitry.
Better Audio. The new circuit has a dedicated low-power supply
Smaller, and neater form factor. The A+ board is about 2cm shorter than the A, and has had the composite video moved to a 3.5mm four pin jack.
If you're aiming for a low powered device that needs to fit in a smaller space, or you need a hat (because who doesn't love hats?), the A+ is probably the better bet.
Model A:
Model A+: