They’re almost interchangeable – but there’s a difference of emphasis that can be useful.
If you talk about the typeface, your focus is on the end result, some type’s appearance and aesthetics in use. It might have come from a font, or it might not: hand-painted signs, graffiti art, comic lettering, calligraphy, logos etc can all have distinctive typefaces without fonts.
If you talk about the font, your focus is more on the product, the item or package that can be bought, downloaded or stored in a box, etc. That font is usually a package for a typeface, but not always: Wingdings, Chartwell and icon fonts like Font Awesome are fonts without typefaces.
Here’s an analogy I adapted from this Fontfeed article, "Font or Typeface?":
Use "typeface" when you’d use "song" (e.g. "I love that
song/typeface …"), and "font" when you’d use "track" ("… so I’m
going to buy the track/font for it").
Most of the time, people use "font" and "typeface" interchangeably, but occasionally you need to focus on one or the other, like how sometimes musicians write great songs, but release bad recordings of those songs or never record them.
Lots of type foundries produce amazing typefaces that make for frustrating fonts because they lack important glyphs e.g. for international use or their hinting is flakey at certain sizes or their default kerning tables are inconsistent, etc etc.
People might say that Arial is a poor typeface (derivative and uninspiring), but a valuable font (huge range of glyphs, great international support, reliable at all sizes and on all devices, etc etc).
It’s a subtle difference, but often a useful one.