Well, I guess I should admit right away that I am biased. I have developed in MS Access for over ten years while I worked with FileMaker for a few years for a client project.
There has been quite an argument between the PRO-FileMaker crowd and the PRO-Access crowd.
So, Which one is better? In my experience, that depends on what is needed and the level of experience of the person developing the database. FileMaker is geared toward people with no programming experience and is extremely easy to get started with. I personally find it to be somewhat limited in its scripting language, but is a great tool for many users. MS Access includes Visual Basic for Applications (VBA), a full programming language that gives the programmer a great deal of control.
While FileMaker is great for the database novice, I would use MS Access for any serious desktop database development.
Regardless of which desktop database system you use, any database that is not properly designed (relationships, normalization etc.) is likely doomed to failure.
Which one do you prefer and why?