These maps can never be equal unless your manifold has dimension zero.
This has a 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 $d{\exp^M}$ takes values in a neighborhood of $0$ in $ T_{\exp(v)}M$, while $\exp^{TM}$ takes values near $v$ in $TM$. Since $\exp(v)$ tends to be different in $M$ from the basepoint of $v$, these have no chance of generic equality.