How I Use It
As a teacher, I use Dropbox to keep ALL my files. That way, I have access to my school files from home (on my PC) and at school (on my laptop). Since I have installed Dropbox on both, all the files are installed locally.
For my students, I use Dropbox to share files, especially big ones. I use Google Drive for data collection through forms but I use Dropbox to share actual files. It is an easy way for them to submit assignments that can't be submitted via a Google form (i.e. presentations, word and excel docs, PDFs, etc.). I don't use it much for student collaboration but it definitely served a purpose for students to access files from homes and also drop files to me without having to e-mail me directly as some of my students don't have e-mail.
As a teaching tool, Dropbox serves a purpose, albeit, a limited one. It allows students to "drop" files in shared folders with other students and teachers. It also allows teachers to put files in these shared folders that they want students to have access to. This can include class information and assignments, but most importantly, it's a great place to put large files that you normally can't e-mail or share.
But that's where the functionality of Dropbox for students ends. It probably is a better idea to use a Wiki tool for sharing files and collaboration. But if a Wiki is too time consuming and doesn't meet your needs, using Dropbox to share and accept files is a great alternative to use with students.
Now, the beauty of Dropbox for me as a teacher is the ability to keep all my files in Dropbox folders and have them installed locally in every PC, laptop, or tablet (not entirely stored locally but still accessible). This is a huge advantage because if you lose internet access, you can STILL access the local files. I have instant back-up of everything I do. As an online and local storage space for my school files, it's perfect. As a collaborative tool for students, it's just OK.