Coming to the Moving rate, Since ion strength of TBE is higher than TAE, solutes in TBE move faster than in TAE. Moreover, temperature of gel increases using TAE-gel for a long time, pH might significantly decrease because of Tris's temperature dependency of pKa. Fluorescent moiety of labelled DNA might go bad in the conditions of high temperature and lowered pH.
And for the Buffering capacity of TBE is greater than that of TAE.