What's the difference between around the clock and 24/7?

As the comments already suggest, there is a subtle difference, which is how long they imply an action goes on for.

Around the clock means without ceasing, even during the night when most normal activity ceases. It can (and usually does) mean that one or people stayed up all night, but can also mean a larger collection (say, the workforce of a factory or the staff at a consulate or legal firm) between them worked “all hours”. It may not be completely literal (factories may have short pauses during shift changes, for example; lawyers may order coffee).

24/7 (as @user121863 and @weather-vane pointed out) has a stronger connotation that not only does the activity continue all day, every day [of the week] – implying for more than a week – but (by usually not mentioning an individual week) implies that this is the pattern every week. If a time period is mentioned, it indicates that the pattern (no hour excluded, no day excluded) continued throughout that period.

In normal English usage, both mean non-stop, but typically the first is used for worked without stopping to meet a deadline, and the second for the lights never went out; the people and the machines kept moving hour after hour, day after day….