C function error: is it better to abort the function or simply exit the program?

The title says it all. Since C doesn't have exceptions, I'm not exactly sure how to handle errors. I've thought of the advantages and disadvantages of both:

ABORTING: Basically what I mean by this is to return an error code (which will be declared in a .h file, maybe with its own perror()-like function) and abort the function, and the obvious advantage is that it helps the user do error-handling, but the disadvantages are:

If the function is not checked every time after it is executed for an error, and an error does indeed occur, it could cause big problems as the program progresses and the user will have a hard time finding where the issue is coming from.
Looking through the header file for the error codes can be tough and annoying.
Error codes may conflict with error codes in other libraries or the builtin C error codes.

EXIT THE PROGRAM: Pretty self-explanatory: as soon as an error is found, print the error to stderr and exit. The advantage of this is that if the error message is detailed enough, the user will easily know what is wrong with their code and fix it, but the main disadvantage is that the user will not be able to write any code that could handle a possible error and would instead have to change the code itself (and it becomes a bigger problem when you need to ask for input or something similar, in which there are millions of possible errors that could arise from incorrect input).

Solution

This largely depends on what your program is doing. Some programs such as simple command line utilities will just abort on invalid input for example, without affecting much neither the user experience, nor the system stability. The user will simply correct themselves and rerun. On the other hand, if it is a safety-critical system, such as military, medical or transportation equipment (read autopilot, pacemaker and such) aborting its program will cost human lives. Or as suggested in the comments - a simple word processor. The user might be very unsatisfied if they loose all their work after made some stupid mistake which caused some program error.

So the general approach to writing a robust software would be to classify your errors as fatal and non-fatal. Non-fatal are the ones you can anticipate during normal program run and can be gracefully handled in a way which allows the program to continue. Fatal ones are the ones which are caused by some abnormal conditions (hardware failure, missing components and such) which make the program not to be able to continue.

Depending on the system nature you might want to loosen or tighten the above classification.