I somewhat echo the previous poster's sentiments with the additional observation that Emory and UT offer different kinds of Public Law specialists. Generally, as I understand things, public law breaks down into studies of "judicial politics" and "law and courts." Judicial Politics are generally associated with traditional behavioralist studies of American Politics (highly quantitiative and rat choice)... think scholars like Segal (Stony Brook), Epstein (WashU), Laurence Baum (OSU) and Neal Tate (Vanderbilt). These folks study judicial behavior in the same way scholars like Fenno study Congress. Then you have Law and Courts folks who are often involved in historical approaches and favor interpretive methodologies that fit well within an American Political Development tradition. Howard Gillman is probably the father of this movement, but today there are strong APD/Law and Courts/Constitutional Development Studies faculty at UT-Austin, Boston College (Ken Kersch, Shep Melnick, Mark Landy), Princeton (Keith Whittington).
I would say re: Emory vs. UT that Emory falls more in line with the Judicial Politics behavioralists whereas UT offers both with probably a stronger flavor of Law and Courts scholars who incorporate qualitative and historical approaches to their study of the court as an institution.
It basically comes down to whether you want to study the court as an institution (its development, role and historical place in American Politics) or study judges themselves (their behavior, their rat choice preferences, and outcomes based jurisprudence)....