This is the fundamental difference:
AC capacitor does not have polarity ...
Described as "non-polarized", they can handle a voltage of either polarity on the terminals and, thus a an alternating voltage. DC is just a special case where the polarity of the voltage does not change, so you can use AC capacitors - as is - in a DC application.
NOW comes the big warning .... when it comes to capacitors used in MAINS circuits, you must be very careful to get properly rated capacitors. A prime example is the X and Y rated capacitors used for EMI suppression. DO NOT get this wrong.
Just as an extra fun fact ... you can make a non-polarized capacitor by taking two polarized capacitors and connecting them in series - back to back. Whether that's a good idea or not is another thing.