As Dave has already discussed, some aircraft use their registration number (such as
SE-XYZ) to identify themselves. Since any one aircraft will have only one registration number, but can fly any number of routes, it clearly follows that a single call sign can be used over time to identify the same aircraft flying different routes, in the case that the callsign identifies the specific aircraft. This is almost certain to be the case in non-scheduled aviation.
Also, specifically in radio communications, it's common for aircraft registration numbers to be abbreviated when used as call signs. Thus you might end up with
S-YZ for the above examples. Worse yet, the aircraft that was
S-YZ a few minutes ago might now be
SE-KYZ entered the same airspace using its registration number as its call sign, causing a collision between the abbreviated call signs because both would abbreviate to
However, there's another case, which is perhaps more common in commercial, scheduled aviation. Simply enough: diversions. Aircraft sometimes have to land on an airport other than the intended one, for any of a multitude of reasons, not the least of which being weather conditions at their intended destination. Depending on circumstances, this could be an airport that they fly over or near on their normal route, or it could be an airport quite some distance from their originally planned flightpath.
Changing callsigns due to diversions could potentially cause confusion for air traffic controllers, in a situation where there's already some other situation causing increased workload (if one flight has to divert, it's possible others will have to, too), so doing so would likely increase the risk of mistakes. Since aviation in general is very risk-averse, it makes sense to avoid the unnecessary risk and just keep using the same callsign.
A similar reasoning can be applied also to flight numbers as seen by travellers; nobody wants the jarring experience of boarding Acme Airlines Flight 1234 and disembarking Acme Airlines Flight 9876 at the end of the hop! It's bad enough that you ended up somewhere other than you planned.
Thus we have shown that two major categories of air traffic, namely unscheduled (commercial and non-commercial) and scheduled commercial (which in turn can be passenger, cargo or both), can or do keep using their original call sign even if at least the destination airport changes. Consequently, the answer to the question in the title,
Do flights of a specific callsign always depart and arrive at the same airports?
is clearly no because while the departure airport may be the same for any given callsign, the destination airport can change in-flight without the callsign changing, meaning there is no one-to-one mapping between the two.