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.
But what does it all mean? Well, it all boils down to this: digital images are made up of numbers, and numbers need context. When converting a profile, you honour the context, and change the colour accurately; when assigning a profile, you ignore the context, and you get an unexpected result.
Consider this. I'm an Aussie, and we discuss distance in kilometres. Let's say I'm visiting the USA, and a friendly local directs me to the nearest motel: "Drive eight miles and take a left - you can't miss it!". What would happen if I simply traveled eight kilometres and turned left - I'd be lost, right? I can't just turn eight miles into eight kilometres and expect to end up in the right place, can I? I have to do a bit of calculation to deduce that eight miles is almost thirteen kilometres ... then I can find my motel.
I could suggest endless analogies for this ... losing half a pound of weight is good - losing half a kilogram is better! It's all about the unit, not just the number.
You'll be well aware that digital images are made up of three channels, each ranging from 0 (dark) to 255 (bright). But on their own, those 0-255 numbers don't mean anything. To interpret, display and print those numbers properly, the numbers need context - namely, colour space. All colour spaces are numerically represented by 0-255, but 255 red in the ProPhotoRGB space is significantly more vivid than 255 red in the sRGB space, for example. The numbers are the same, but the context is different.
If you assign a profile, it's like replacing "miles" with "kilometres" and ending up in a different place. Assigning a profile keeps exactly the same 0-255 numbers, but changes the colour space to which they belong. Result? A different colour. Not cool. (A variation of this occurs when you view your web images in a non-colour-managed browser and see a difference.)
If you convert a profile, it's like converting 8 miles to thirteen kilometres, and ending up safely at your destination. Converting a profile changes the numbers as well as the colour space, to achieve the result you expect (apart from any channel clipping which may occur in the conversion).
Profile conversion is the engine which drives colour management. There are profile conversions happening all the time behind the scenes to ensure that the correct numbers are showing us the correct colours. If you're a nerd, it's fascinating! If you're not, it's sleep-inducing. Luckily, the wonderful nerds at Adobe have taken care of most of it for us. We don't need to manually change profiles very often at all. But if you do - for printing, usually - make sure you convert, don't assign.