Confused? Well this question gets asked a lot:
What's the difference?
Well here's my take.
If you have a photo with or without an associated profile you can embed a profile though the assign/convert dialogues, Image> Mode>
this is the assign dialogue box.
REMEMBER: It is important to preserve your embedded profile!
You would use the Assign command to remove a profile (that was suspect) to select an alternative, or select a profile for an
'untagged' image. I rarely use this unless the image is 'untagged' or the embedded profile is way off (probably incorrectly embedded by someone).
My advice for beginners is not to use assign unless you know the file has no profile.
You can also use the Convert dialogue to place a new profile in a image that is tagged or untagged.
The source is normally your working space, and the destination is the space you need to convert to.
It's best to use the Adobe conversion engine, the rendering intent is up to you, I use Relative.
What's the difference?
Well the difference is how they treat your image; Assign leaves your pixel values unchanged so the image will look different.
Convert does change the pixel colour values and uses the colour engine to maintain the appearance of the photo.
Generally I assign a profile if I know the image has no profile and convert when I save an image for the Web.