In the beginnings of the internet there was only two options for animation on a webpage. The first was animated gifs, an animated gif is essentially a web-based stack a flip cards, stringing together static images to create the illusion of movement. Unfortunately animated gifs can end up being large files, pixilated and don't look very sharp. In 1996, Macromedia released a product named Macromedia Flash. This was a vector based animation platform that allowed web designers to create animations using tools provided by Macromedia.
jQuery or Flash
Changes to the design of jQuery features most front end designers can handle and this will lower any maintenance costs you may incur. A Flash application encapsulates all the code in the application itself. First, this means creating the application require the use a specialized Flash developer, but this can be to your advantage if you are looking for features to your site such as audio/video player or 3D animation. If you are looking to have complex multi-media in your project, you are going to have a higher development cost. Changes and updates to Flash content can be executed via updateable XML files, but if you want to change the design of the application you will need to use a Flash developer with can cost more in maintenance hours. What this decision really comes down to is asking yourself what are the features that you need. Much of the "flash" in Flash can be mimicked at a lower cost with jQuery, but if your site is multi-media heavy; your best option would be to go with the Flash application for the ease of the end user and yourself.