This ($t=1$) is a particular case of the famous Catalan equation. The only solution known is $3^2=2^3+1$. The case $p=3$ was done by Euler and the case $p>3$ was done by Ko Chao in 1964 (the English proof is published in Mordell's book); E. Chein published an elementary (and very nice!) proof of Chao's theorem in 1976 (PAMS **56**, pp. 83-84).

The general $t\in\mathbb Z\setminus\{0\}$ (fixed) case is a special case of Pillai's conjecture: it is expected that only finitely many solutions in integers show up.