This is a great example of why the OPs question could be a good or terrible interview question -- it's not clear if they want the obvious technical solution (comparing datetimes) or a wider discussion about "what is a date time and how is the data represented", "what are the real world / business implications", "here is existing technology using intervals that will solve it" etc as you mentioned.
In my experience interviewers are usually looking for the technical solution despite the business oriented solution usually being much more applicable (and thus relevant) in the day to day role.
Key thing to remember here as an interview candidate is to clarify with the interviewer the scope of the question and the nature of the answer they're looking for. If for instance the interviewer starts with the simple technical solution and then probes the business aspects this might be a nicely rounded question.
My company asks a question during engineering interviews that is overly simple, but a bit vaguely defined on purpose. The main objective is to see if they can ask clarification questions to determine precise requirements, which is something that every engineer does daily on the job.
Can I assume the times are in a sane date format/datatype? If not, start with e1 = toSaneDateFormat(endtime1), s2 = toSaneDateFormat(startime2) (Fill in if interviewer is interested).
Then, as you say, a check for overlap is easy - but maybe one wants to be more fancy, like: if (timeDelta(e1, t2) < timeToWalkFromAtoB, or < 5 minutes -- they should be considered an overlap? If they are on different continents, maybe < 24 hours should be an overlap?
At any rate, I'm guessing (hoping) this leads to discussions about representing dates, and what the business logic is (eg: physical meetings - you can't teleport from one location to another).
[ed: And as others have touched on, if you deal with timestamps/raw number types - be careful that you don't end up with appointments in "wrong" order - I'd say a sort() aware of date-objects might be your friend here.
ed2: In fact, if you can assume a sane date-type, and timeDelta, you could probably assume an "interval" type, and simply ask for overlap?(appointment1, appointment2) ... ]