Is it possible that solid state drives (or any faster drive) will make common applications faster even if they are cached?

By default, Firefox caches all downloaded content to disk. You can set Firefox to use your RAM as the cache, although I am unsure wether or not newer versions of Firefox have this option enabled by default. For your reference, this is much different then changing the memory cleanup and garbage collection options, as you can force Firefox to use more RAM and use less disk cache - which is one option I highly recommend if you use a solid state drive.

Before I go into how you can do this, note that if you have the choice between using a disk drive (mechanical or solid state) or RAM for cache, regardless of the application, go with RAM always (that's what it's there for!). Not only is it quicker and with less access time, but DRAM does not have a limited number of writes as opposed to SSDs. Again though, this assumes that you have adequate memory to use as a cache in the first place - see the bottom of my answer for more about this.

Now, to set the amount of RAM Firefox uses before using disk cache, navigate your browser to about:config. In the Filter, enter "memory", and you should see:

browser.cache.memory.capacity

If you don't, then you will need to manually create this key as an integer value. This integer value represents the memory cache (in kilobytes) that Firefox can use. I personally set this value to 393216, or 384mb. To avoid Firefox running out of memory, you can then increase the disk cache it uses by changing the following variable (you can just use "disk" as the search filter):

browser.cache.disk.capacity

Again, this is the disk cache (in kilobytes) as an integer value, and if it does not exist, you must create it. I personally set mine to 0 after switching to a solid-state drive, although you can set this to whatever you want (IIRC, the default value was 300mb or 307200). Note that this is different from the amount of storage Firefox uses to store offline files or your web history.

Now, back to using memory versus a disk drive as an application's cache. For a solid-state drive, assuming that you have purchased one, it is (usually) assumed that your system has adequate memory (otherwise you wouldn't have spent 10x more on a low capacity drive). Due to the limited write cycles of solid-state drives, you should also disable any pagefiles/swapfiles on SSDs, since this will cause an artifically higher number of writes to the drive, decreasing its lifetime. Disable your pagefile/swapfile to force the system to use more memory. If you still want a safe buffer, or still need virtual memory, then move the pagefile/swapfile to a hard disk drive.

Yes, it may be slower, but at least you won't burn out your shiny new SSD any faster. This point should make those on a tight budget question whether or not they should allocate their funds to more RAM or a (bigger) SSD. The choice depends on your needs, but keep these points in mind.

Solid state drives should never be used for caching for the limited lifecycle reason - that's what your RAM is for, and most application developers (should) write programs keeping this in mind. Regardless of the speed of SSDs vs HDDs, they still present a bottleneck compared to your memory bus bandwidth.

Well, of course it would make a difference to Firefox, or any other browser, as they use browser caches, where they look for files which they have already downloaded, when loading new pages (eg. the Superuser logo on this page and the site's CSS files), although some of this is most probably cached in RAM as well.