Ashkenazi and other Jews have a direct link to the Land of Israel. This link is an integral and an essential part of out identity and it has been kept within us for generations.
Khazars, on the other hand, were a Turkic people, who were pagans most of the time, but briefly (for about a 100 years) adopted some form of Judaism as their official religion. It is still a matter of debate among historians whether all of them converted or just the nobility. Anyway, the Khazar state was destroyed and Khazars disappeared as a people from the face of the earth in 10th century, more than a 1000 years ago.
The whole “Khazar story” is nothing but an anti-Zionist myth.