A typeface is the design of the letters (e.g. Helvetica); a font is the computer file containing that design in a specific style (e.g. Helvetica Bold as a .woff2 file). A typeface is designed; a font is installed.
The distinction may sound pedantic, but it has practical – especially licensing – consequences. When you "buy a font for the logo", you buy a licence for font files, and its scope (print, web, app, pageviews) decides where you may use the typeface.
Worth knowing: a desktop licence usually does not cover embedding the font on a website; webfont licences exist for that. When choosing a brand typeface we check not only aesthetics but available styles, language support and licensing costs across all planned media.
Trivia: historically a "font" was a set of metal type in one size and style – digital usage inherited the word.