I use both.
Dropbox is basically just cloud based storage for files, which conveniently syncs between multiple devices if you wish. I have my personal documents on Dropbox as well as all my processed photos. That way they're backed up off site and readily available on my two laptops, my phone and all over the world. Think of it like off site networked storage, with a cached and synced copy on your local drive. You can stick any sort of file into Dropbox, so long as it meets storage constraints for your account.
One cool feature I like is that when I snap photos on my smartphone Dropbox will upload them to the cloud (if I wish) and once I'm at my laptop they immediately sync from the cloud to my hard drive and are ready to view/edit within seconds, and all without lifting a finger.
Evernote, as John says, stores "information". I use it to hold reference information such as software licence keys, travel details etc. and screen scraped captures of information from websites. From a mobile phone you can record photos into Evernote, record some audio, draw a picture and write notes to go with these things. It will also sync across devices.