The difference between assigning a profile and converting to a profile is one that confuses a lot of newcomers to colour management. I have pondered whether or not it's possible to come up with a concise explanation that does help at the risk of over-simplifying, omitting necessary detail and leaving the reader just as confused as before. Anyway, here's my stab at it.

Just as there is no colour in the physical world (colour is just the way our eyes and brain interpret the different wavelengths of light falling on the retina), there is no colour in our digital images. There are only numbers representing colours. I'll use RGB to illustrate this, but the same concept applies to CMYK, or LAB. Every pixel in a converted 10D image has three numbers indicating the amount of red, green and blue. Different coloured objects in the pictures will have different RGB values for their colours. If I take the same picture with a different camera, the RGB values may well be different because the two camera "see" colour differently. We use colour spaces or profiles as the reference framework for interpreting colour numbers.

I think language can be used as a helpful analogy. Take the words: rouge, vert, bleu. If you have never been given any instruction in French, you won't recognise these words or have any idea what they mean, but if you're told they are French you can get a French-English dictionary and work out their meanings.

Devices like digicams have "dictionaries" - we call them profiles or sometimes colour spaces. Output devices like monitors and printers also have profiles. A profile "describes the colour reproduction behaviour of a device" (Bruce Fraser et al.). Using a profile ensure consistent colour in our workflow, so one of the first steps in working with a digital images is to assign the correct profile. This can be done in Photoshop: