As far as I know, there would be no way for scripts to directly access the data in a Bento database, so you would need to manually export that data every time you need to update the website. If you don't have too many sets of data, or if the site doesn't need to be updated too frequently, this might work well for you.
Also, Bento is fairly limited when it comes to building relational databases. It can handle flat databases and very simple relationships, but lacks the ability to set up complex relationships between tables. If you don't need complex relationships, and manually exporting your data is acceptable, then Bento would be a viable option that's relatively inexpensive and fairly easy to learn and use.
On the other hand, FileMaker Pro is very scriptable, allowing for almost complete control through both its own internal scripting language as well as through AppleScript. It also handles much more complex relationships. Of course, this advanced feature set also comes with a higher learning curve (and price tag). However, FileMaker does have a large and active community of users that are very helpful, and there are plenty of tutorials and sample databases to help get you up to speed.
While FileMaker does have built-in web publishing, there is a limit of 5 concurrent users unless you get FileMaker Server, or host your database on a 3rd party FileMaker hosting service.