An A/D converter is the specific piece of hardware that takes an analog signal and turns it into its digital representation.
A box or card that has maybe jacks for analog in and out and A/D converters for them, maybe S/PDIF or ADAT optical or AES/EBU digital port, maybe a MIDI connector, maybe a Word Clock port and such, and has drivers and mixer applets so that the analog signal ends up inside the computer to be manipulated... this is an audio interface.
So an A/D converter is often one element in an audio interface. Though I guess it's possible to have a card that accepts digital audio inputs only and call it an "audio interface" too. You also might call a stand-alone box like an Apogee, which is designed to just do one job --converting analog signals to digital data -- an A/D converter, but I guess technically if you have jacks to put signals in and jacks to take digital data out, it's an "audio interface" too, really.
It boils down to semantics and marketing. If an audio interface does only the job of converting an audio signal into digital data, and providing a path in and a path out, then it's typically called an A/D converter. If it does this job as well as get the data into the computer to be handled, and back out again to be heard, it's usually called an "audio interface." Often when it's all on one card -- and often when it's not, too -- it's called a "sound card."
Hope that helps a bit...
