Assigning a profile is
not converting the profile embedded in an image.
An image coming from a raw converter will have an embedded profile, usually sRGB or Adobe RGB, depending on the options you selected in the raw converter. Let's say, for the sake of argument, the embedded profile in the image from the raw converter is the Adobe RGB colour space.
If you assign a profile, let's say sRGB, to the aforementioned image, you end up with a file that uses the Adobe RGB colour space but which is identified by software like Photoshop as an sRGB image. In other words, you have a mismatch between the colours actually in the image (Adobe RGB) and the instructions on how to display those colours (sRGB).
There are only two instances when you should assign a profile, both uncommon: (1) when a profile is missing (e.g. most images without profiles are generally sRGB - but you might want to assign the sRGB profile so that any colour-managed program will know it's an sRGB image); (2) when a file appears to have a wrong file (like the aforementioned Adobe RGB image with an sRGB identifier - which now needs the Adobe RGB profile assigned, to replace the wrong sRGB one with which it's flagged).
As fbf says, if you want to change the profile in an image, what you should be doing is
converting the profile, using Photoshop's "Convert to profile" function. This both attaches the new profile and converts the image colours to the new profile.
When you convert from the Adobe RGB colour space to the sRGB colour space, you won't usually notice much, if any, change, as most colours are unaffected.
The Adobe RGB is larger than the sRGB colour space - in particular, it contains more bright greens and blues: so, if your Adobe RGB image has a lot of turquoise, say, you will see more of change when you convert it to sRGB, because these colours don't exist in the sRGB colour space and must therefore be approximated colours that do exist.