In short there is no difference; they are simply differing terms used to describe the same generic product category.
The term sound card is certainly the older of the two and developed from products (e.g. Yamaha SW1000XG, or the DB50XG daughter board) wherein it was not uncommon for sound cards to actually contain wavetable synths as well as offering audio input/ output paths to and from a PC. In the early years of PC based recording, there was also no such thing as USB or Firewire and all recording was done via installing some form of PCI card, or even ISA, within a PC. This is probably how the term evolved.
However, these days USB and Firewire have led to significant advances in both performance and the feature set found on audio interfaces servicing this particular market. Wherein the old days, the line inputs and outputs of a sound card were generally connected to an external mixer, which provided the necessary microphone pre-amps and routing, this complex set-up has somewhat been replaced by one box solutions (e.g. Focusrite Saffire Pro 26, Presonus Firepod & Tascam US144), hence the term Audio Interface.
Generally these days, we tend to view Sound Cards as PCI products with line inputs and line outputs, whereas any product that offers microphone pre-amps and headphone monitoring is considered an audio interface.