Brett Kelly, author of the popular Evernote Essentials guide, writes about how he uses both Evernote and Dropbox for different purposes:
This isn’t to say that, should one be so inclined, somebody couldn’t make Evernote into a kind of a Dropbox or Dropbox into a kind of an Evernote. The similarities are strong enough (barely) such that one could theoretically be used in place of the other. I’m not a fan of this idea because I’m a firm believer in using the right tool for the job.
I have to agree. I am also a user (and fan!) of both services, and happily pay for both of them each month ($5/mo for Evernote Premium, and $10/mo for Dropbox Pro 50GB, although I might even upgrade the Dropbox account to the 100GB soon).
Each service was clearly intended for different use-cases. For example, my Dropbox is essentially a replacement for the 'My Documents' folder on any PC I use. It's where I store my iTunes library, including several movies for streaming on my iPhone via the mobile app, as well as all of my personal photos, installers for many useful applications, various documents (sorted by type), backups of files I may need later, and more.
My Evernote account is mainly used to store 'clipped' content from websites, or notes that take advantage of Evernote's multimedia features. Examples would include recipes, gift ideas, stuff I want to buy later, travel notes from cities that I've visited, important information that I may need for my full-time job (yes, I have a job outside of Unretrofied; you think I actually make money doing this?), copies of receipts, stuff I have liked (photos, wines, foods, art, etc), how-to articles for projects, blog post drafts, various sorts of inspiration (style, fitness, design, writing, etc)....the list goes on and on and on.
I agree that Dropbox could feasibly be used for the same purposes that I use Evernote for, but it would be far more convoluted. I would rather just highlight content on a webpage and click the 'Clip to Evernote' button in my browser than try to recreate a webpage layout in a Word or PDF document. Sometimes you don't even have to do that much! If you just click the Clip button without highlighting anything, it can usually determine what content you're looking to save automatically (automagically?) except in cases of weird website layouts.
The nice part of Evernote is that everything is easily tagged and searchable, including the text within images. Dropbox has no such feature, you can only put certain files in certain folders for your own organizational purposes, which I have done since I'm borderline OCD, but it's just not the same.
Evernote is a glorified bookmarking service, while Dropbox is a place to store files (although I do occasionally attach files to certain notes within Evernote). There's room for both in the world, and I don't see either of them really reducing the value of the other.
