We report a large tunneling anisotropic magnetoresistance (TAMR) in (Ga,Mn)As lateral nanoconstrictions. Unlike previously reported tunneling magnetoresistance effects in nanocontacts, the TAMR does not require noncollinear magnetization on either side of the constriction. The nature of the effect is established by a direct comparison of its phenomenology with that of normal anisotropic magnetoresistance (AMR) measured in the same lateral geometry. The direct link we establish between the TAMR and AMR indicates that TAMR may be observable in other materials showing room temperature AMR and demonstrates that the physics of nanoconstriction magnetoresistive devices can be much richer than previously thought.