Not sure you use either of those terms in the context of a CDS
For bonds -
Credit spread is spread above treasury curve where (ie bond yield - gov bond yield) where the treasury curve is interpolated for bond maturity
Asset Swap Spread is a bit more complicated. Conceptually it's an exchange of the bond's (fixed) cash flows for the floating leg of a swap. ie how much do I have to vertically bump my swap curve so that the net present value of my floating payments sum to 100.
Typically there's not that much difference between a credit spread and the asw spread, but certainly the asw is deemed to be more accutare.