They're right next to each other in Photoshop's "Edit" menu, and they both involve changing your image from one colour space to another (eg Adobe RGB to sRGB); but they're vastly different functions, and definitely not interchangeable.
Let's be clear up front - if you need an image to go from one colour profile to another, it's 99.99999999999% likely that Convert to profile is the one you need. Assigning a profile is so darn rare that I can't think of a single reason why a photographer would need to do it in their day-to-day workflow.
Even if you read no further, this is the important message - Converting is your friend; assigning is almost always your foe.