I've been doing some research using flight data, but I'm not particularly versed with aviation. From a quick look, it appears that, say, a flight with callsign SAA203 will always be departing from Johannesburg, South Africa and arrive in JFK Int'l airport.
Is this sort of thing always the case, are callsigns only changed when a flightpath is retired, or can a given callsign correspond to multiple flightpaths?
As Dave has already discussed, some aircraft use their registration number (such as N12345 or G-ABCD or 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 2345, G-CD or S-YZ for the above examples. Worse yet, the aircraft that was S-YZ a few minutes ago might now be S-XYZ because 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 S-YZ!
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.
You may be confusing callsign (with can be multiple things) and flight code/number as FlightAware may display one or both of these things depending on whats applicable. FlightAware tracks publicly available filed flights at a variety of levels depending on what callsign you search/are looking at the answer may change. For example
American Airlines runs flight AA66 (AAL66) from JFK to BCN and the code AA66 is used to identify that flight (AA67 is the return flight). This code may also be used for a code share. Generally these codes stay fairly consistent unless there is an overlap (see ymb1's) comment for how dupes are handled.
Generally non scheduled commercial and private traffic the flight files with their tail number, for example in the US tail numbers are often 5 (or possibly 4) digits and a preceding N. So N12345 may file for a flight from anywhere to anywhere they want to go. This history log is a good example of a charter plane that seems to fly out of KDYL to lots of places.
No, not necessarily.
In the case of the air carrier that I fly for, our call signs (and flight number) are specific to each aircraft, regardless of route. Each aircraft in our fleet has an assigned numerical designator—typically derived from the aircraft registration (tailnumber)—which is used with our company telephony designator to create a unique call sign for each flight.
Suppose our company used the call sign "Acme" as a telephony designator with a three letter designator of ACM. And suppose that I was assigned a flight in an aircraft with the registration N123AB. In that case, regardless of route, our call sign for that flight would be "Acme 123" with the coded designator of ACM123.
However, that is just one example of how call signs are designated. Many air carriers—especially airlines that fly scheduled routes—do things differently.
Similar to what Dave says in his answer's middle paragraph, Scheduled Air Carrier (commercial airlines) flights (with a few exceptions) use the same Flight Number. For example, a United Airlines (UAL) flight that is scheduled to depart Denver for a flight to Chicago at 9 a.m. will almost always use the same Flight Number.
However, on some occasions, as an example, UAL may use the same Flight Number Monday through Saturday, but use a different Flight Number on Sunday. But more often, for a flight to to/from major cities where the pax demand is constant and connecting schedules are the consistent, the same Flight Number is used.
Flight numbers on a few routes may change occasionally based on seasonal adjustments (for weather, pax demand etc). However, the change is structured so that it will be published well in advance of the actual change. Also, when there is a major routing/city-pair adjustment to the airline's route structure, a Flight Number between city pairs may change (likely connecting flights, routes, and interdependent schedules will also have changed)
When flights are scheduled between particular city-pairs the Flight Numbers are consistent for many reasons. Importantly, passengers book flights months ahead of time and changing Flight Numbers in an ad hoc manner would be extremely problematic (think of all the connecting and interdependent flights worldwide that could be adversely affected). Also, Flight Numbers and associated routes (city-pairs) are published in paper form, electronic databases, etc. months ahead of time. Further, airline yield management (economics) demands a sophisticated routing structure where the Flight Number necessarily serves as an integral component internally and with code-share partners worldwide.
The flight numbering system for scheduled Air Carriers is far more complex and structured than it may appear to the uninitiated. Pre-planned/consistent flight numbering and the city-pair relationship with that numbering is essential to a successful marketing schema.
Lastly, I'm certain that some who read my comments will come up with examples showing something different than what I have said. But, while there are some exceptions, I believe a careful reading and analysis of Scheduled Air Carrier schedules with associated city-pair flight numbering will support my answer.
