(The differences go beyond their coats--they're separate terrier types.)
Here's a wire fox terrier, below, in an ad that totally baffles me. I can't translate the ad's references.
Can you?
Are those falconer gloves the woman wears? Why golf clubs? Is that her lover, a Columbian generalissimo, in the train? What's with the fire from above?
Here's what I've come up with:
"Weetabix Will Keep You Perky, Even During an Alien Invasion."

(Ad via The Wire Fox Terrier)
Anyway, here's a smooth fox terrier.

They aren't as famous as the wire foxes, but perhaps that's changing, after director Tom Ford cast his own SFTs in A Single Man (2009).
Here they are with partners George ( Colin Firth) and Jim (Matthew Goode). (From the blog Raleigh Pop: "Tom Ford's Fox Terriers")

Warning: doggie tragedy ensues. But it's a pretty good film, if visually a bit too much like a Calvin Klein ad.