They're almost interchangeable - it's more a difference of emphasis than a hard difference in meaning.
If you talk about the typeface, your focus is on the end result, the appearance and aesthetics in this particular case. If you talk about the font, your focus is more on the product, the item that can be bought or downloaded.
Here's an analogy I adapted from this Fontfeed article, "Font or Typeface?" which I used in a comment a while ago that seemed to go down well:
Use "typeface" when you'd use "song" (e.g. "I love that
song/typeface..."), and "font" when you'd use "MP3" ("...so I'm
going to buy the MP3/font for it").
If you say "I don't like this font" or "I don't like this MP3 (or "I don't like this track"), people will usually take it as the same as "I don't like this typeface" or "I don't like this song". But you could say, "I love this typeface but I don't like the font - it has no Cyrillic letters and a limited range of glyphs", like you could say "I love this song but I don't like this MP3 / track, the sound quality isn't high enough and the stereo balance doesn't feel right".
For example, some people might say that Arial is a poor typeface (derivative and uninspiring), but a valuable font (huge range of glyphs, great international script support, etc etc).
Or, if you say something is a great web font, you're probably talking about the practicalities of using it online - e.g. universality or how well it renders cross-browser and across operating systems. If you say something is a great web typeface, you probably mean its appearance is well suited for websites. There are plenty of Google web fonts that look like promising web typefaces, but render badly making them disappointing web fonts.
Most fonts have one or more typefaces (excluding unusual non-type fonts, like wingdings, chartwell, etc), but not all typefaces have fonts. For example:
"I love that typeface: what font is it?"
"It's the font [-name-], weight 300. Also, extra-tight kerning has been applied"
"And what font is that other typeface?"
"That typeface isn't a font, it's a scan of some handwriting".