What are the differences between the two?
Both has different BoK (Body of knowledge) with some similarities about test analysis, communication requirements, test estimation and tool selection and management. But at the end of the day, both comes to the same objective: 'Certify individuals on the test management process'.
Cost of both is almost same. However, ISTQB provides 3 layered certification: Foundation, Advanced and Expert in test management. Currently in market is has more credibility than ASQ certification (but again certification is just a piece of paper and real judgement of a person is based on his knowledge and decision-making capabilities). ASQ provides 2 layered certification: Quality Engineer and then Test manager. This certification comes with validity of 3 years, unlike ISTQB certifications, that are valid for life-time. Due to the same fact, few organization recognize ASQ better (due to presumption that person got certified recently, so would 'remember' most of the things)