This may seem like a silly question, but as I am going through the documentation for both services it is difficult to disentangle what each does, specifically. From what I've gathered
BlueMix is essentially an all-in-one cloud platform for accessing various IBM Analytics APIs and you can code in various languages, while
Data Science Experience is sort of an RStudio on steroids (it even allows you to use RStudio), where I can process massive data sets using IBM's resources instead of my own, but I can also code things in Python if I want.
For my needs, I will be using several different types of data, including time-series physiological data and natural language text (sounds like I'll likely need Watson). I would like to be able to use TensorFlow for my work as well.