One of the problems with this question is that $\text{Cont}(M,\alpha)$ depends heavily on the choice of contact form $\alpha$, so while the previous answer shows that there is a choice of contact form for which $\text{Cont}(M,\alpha)$ and $\text{Cont}^+(M,\xi)$ can't be homotopy equivalent, it's not immediately clear if this is true for all choices of $\alpha$.

That said, here's a connected example for which the inclusion also fails to be surjective on $\pi_0$. (Attribution: this emerged out of discussions with Hansjoerg Geiges.)

Let $(M,\xi)$ be $T^3$ with its standard contact structure $\xi_0$, and using coordinates $(x,y,\theta)$ on $T^3$, write the standard contact form as

$\alpha_0 = \cos(2\pi\theta) dx + \sin(2\pi\theta) dy$.

Now for any $A \in \text{SL}(2,\mathbb{Z})$, the direct sum of $A$ with the identity defines a linear map on $\mathbb{R}^3$ which descends to $T^3$ as a diffeomorphism $f : T^3 \to T^3$. It is easy to show that for any such map, $f^*\alpha_0$ can be deformed through contact forms to $\alpha_0$, hence by Gray's stability theorem, $f$ is isotopic to a contactomorphism $f_0$.

However, if $f_0$ is a *strict* contactomorphism with respect to $\alpha_0$, then it must preserve the corresponding Reeb vector field, and this is a very strong restriction. It means for instance that the Morse-Bott torus of Reeb orbits at $\{\theta=0\}$ is mapped to another Morse-Bott torus of orbits *with the same period*, and the only such orbits that exist for $\alpha_0$ point in either the same, opposite or an orthogonal direction. With arguments like this one can show that $f_0$ cannot be a strict contactomorphism unless the matrix $A$ is orthogonal... in fact, I believe it must be a fourth root of the identity.

With a little more work one can say something similar for a much larger set of contact forms, using the fact that a strict contactomorphism must always map Reeb orbits to Reeb orbits of the same period. (I'm fairly sure that for a generic choice of contact form, not only are all Reeb orbits nondegenerate but no two of them have the same period.) Unfortunately, I still don't know how to turn this into any statement for *all* contact forms, but the evidence is certainly against $\text{Cont}(M,\alpha) \hookrightarrow \text{Cont}^+(M,\xi)$ ever being bijective on $\pi_0$.