These maps can never be equal unless your manifold has dimension zero.

This has rather trivial reason. If we look at both of these maps on a neighborhood of $0$ in $T_v T_p M$ for some fixed $v\in T_p M$, we see that $dexp^M$ takes values in a neighborhood of $0$ in $T_0 T_{exp(v)}M$, while $exp^{TM}$ takes values near $T_v TM$. Since $exp(v)$ tends to be different in $M$ from the basepoint of $v$, these have no chance of generic equality.