The audio interface IS the soundcard. The term soundcard is a holdout from the days of yore when you had to physically install an audio card inside the pc case in order to have audio - at that time, pc's did NOT normally come with any audio capability and you had to install a soundcard (usually a 'blaster) into a free expansion port. Nowadays, motherboards come standard with onboard audio.
The definition of an audio interface is a device that:
a) converts digital data into analog or digital audio that can be amplified or translated by another digital audio device. The former means that the interface outputs electrical audio impulses that can be put through an amp to be heard through headphones, speakers and other PA systems. The latter, digital audio, is a format of audio that requires further conversion from digital to analog so that it can be amplified. While digital audio is still data, it is a different protocol from that of .mp3, .wav, or .aif files. You need an interface that converts these files into audio.
b) converts analog audio into digital data that your computer can process. In the old analog days, you could consider preamps and their associated circuitry as interfaces when recording into tape, for example. These preamps and recorders would convert analog audio into electrical impulses which would be converted to magnetic energy to be 'imprinted' onto a strip of iron oxide impregnated strip of plastic which we called 'tape'. Nowadays, we record to hard disk or SD cards; the media may have changed but the principle is the same.
Therefore, a soundcard IS an audio interface. The USB or firewire soundcard IS an audio interface, the only difference being that they are external interfaces compared to an internally installed soundcard. Any device that allows a computer to interact with audio becomes an interface... even a USB mic can be considered an interface by virtue of definition, although it is strictly an input only device - except in some models that include audio output for monitoring purposes. (What?! a mic with an output? Try googling the sE usb mics.)