On a compact manifold, the conditions $(d{+}\delta)\alpha=0$ and $\Delta\alpha=0$ are equivalent, but this is not true on noncompact manifolds or manifolds with boundary, and I suspect that this is the situation in the original question.

For example, take the flat metric on $\mathbb{R}^n$ and consider the case of a $1$-form $\alpha$. The (local and global) solutions of $(d{+}\delta)\alpha=0$ are given by $\alpha = da$ where $a$ is any solution to the (determined elliptic) equation $\Delta a = 0$, while the (local and global) solutions of $\Delta a = 0$ are $a = a_1\ dx^1 + \cdots + a_n\ dx^n$, where $\Delta a_i = 0$ for all $i$. Obviously, when $n>1$, there are a lot more solutions of the latter equation than the former.

Moreover, if one wants to specify boundary data that will make the solution unique, then it's clear that one will have to specify boundary data completely differently in the two cases. For example, for solutions of the equation $(d{+}\delta)\alpha=0$ on a domain $D\subset\mathbb{R}^n$ with smooth boundary, specifying the pullback of $\alpha$ to $\partial D$, say, $(\partial D)^*\alpha = \phi$, clearly won't work unless $\phi\in \Omega^1(\partial D)$ satisfies $d\phi=0$ and $[\phi]\in H^1_{dR}(\partial D)$ lies in the image of the restriction mapping $H^1_{dR}(D)\to H^1_{dR}(\partial D)$. On the other hand, for the second equation $\Delta\alpha=0$, the appropriate boundary problem would be to specify the restriction (not the pullback) of $\alpha$ to $\partial D$.

Similar remarks apply for $k$-forms with $k>1$ on a general Riemannian manifold with boundary. This problem is not immediately reducible to the Hodge theorem for compact orientable manifolds without boundary.

On a compact manifold, the conditions $(d{+}\delta)\alpha=0$ and $\Delta\alpha=0$ are equivalent, but this is not true on noncompact manifolds or manifolds with boundary, and I suspect that this is the situation in the original question.

For example, take the flat metric on $\mathbb{R}^n$ and consider the case of a $1$-form $\alpha$. The (local and global) solutions of $(d{+}\delta)\alpha=0$ are given by $\alpha = da$ where $a$ is any solution to the (determined elliptic) equation $\Delta a = 0$, while the (local and global) solutions of $\Delta \alpha = 0$ are $\alpha = \alpha_1\ dx^1 + \cdots + \alpha_n\ dx^n$, where $\Delta \alpha_i = 0$ for all $i$. Obviously, when $n>1$, there are a lot more solutions of the latter equation than the former.

Moreover, if one wants to specify boundary data that will make the solution unique, then it's clear that one will have to specify boundary data completely differently in the two cases. For example, for solutions of the equation $(d{+}\delta)\alpha=0$ on a domain $D\subset\mathbb{R}^n$ with smooth boundary, specifying the pullback of $\alpha$ to $\partial D$, say, $(\partial D)^*\alpha = \phi$, clearly won't work unless $\phi\in \Omega^1(\partial D)$ satisfies $d\phi=0$ and $[\phi]\in H^1_{dR}(\partial D)$ lies in the image of the restriction mapping $H^1_{dR}(D)\to H^1_{dR}(\partial D)$. On the other hand, for the second equation $\Delta\alpha=0$, the appropriate boundary problem would be to specify the restriction (not the pullback) of $\alpha$ to $\partial D$.

Similar remarks apply for $k$-forms with $k>1$ on a general Riemannian manifold with boundary. This problem is not immediately reducible to the Hodge theorem for compact orientable manifolds without boundary.

On a compact manifold, the conditions $(d{+}\delta)\alpha=0$ and $\Delta\alpha=0$ are equivalent, but this is not true on noncompact manifolds or manifolds with boundary, and I suspect that this is the situation in the original question.

For example, take the flat metric on $\mathbb{R}^n$ and consider the case of a $1$-form $\alpha$. The (local and global) solutions of $(d{+}\delta)\alpha=0$ are given by $\alpha = da$ where $a$ is any solution to the (determined elliptic) equation $\Delta a = 0$, while the (local and global) solutions of $\Delta a = 0$ are $a = a_1\ dx^1 + \cdots + a_n\ dx^n$, where $\Delta a_i = 0$ for all $i$. Obviously, when $n>1$, there are a lot more solutions of the latter equation than the former.

Moreover, if one wants to specify boundary data that will make the solution unique, then it's clear that one will have to specify boundary data completely differently in the two cases. For example, for solutions of the equation $(d{+}\delta)\alpha=0$ on a domain $D\subset\mathbb{R}^n$ with smooth boundary, specifying the pullback of $\alpha$ to $\partial D$, say, $(\partial D)^*\alpha = \phi$, clearly won't work unless $\phi\in \Omega^1(\partial D)$ satisfies $d\phi=0$ and $[\phi]=0\in H^1_{dR}(\partial D)$. On the other hand, for the second equation $\Delta\alpha=0$, the appropriate boundary problem would be to specify the restriction (not the pullback) of $\alpha$ to $\partial D$.

Similar remarks apply for $k$-forms with $k>1$ on a general Riemannian manifold with boundary. This problem is not immediately reducible to the Hodge theorem for compact orientable manifolds without boundary.

On a compact manifold, the conditions $(d{+}\delta)\alpha=0$ and $\Delta\alpha=0$ are equivalent, but this is not true on noncompact manifolds or manifolds with boundary, and I suspect that this is the situation in the original question.

For example, take the flat metric on $\mathbb{R}^n$ and consider the case of a $1$-form $\alpha$. The (local and global) solutions of $(d{+}\delta)\alpha=0$ are given by $\alpha = da$ where $a$ is any solution to the (determined elliptic) equation $\Delta a = 0$, while the (local and global) solutions of $\Delta a = 0$ are $a = a_1\ dx^1 + \cdots + a_n\ dx^n$, where $\Delta a_i = 0$ for all $i$. Obviously, when $n>1$, there are a lot more solutions of the latter equation than the former.

Moreover, if one wants to specify boundary data that will make the solution unique, then it's clear that one will have to specify boundary data completely differently in the two cases. For example, for solutions of the equation $(d{+}\delta)\alpha=0$ on a domain $D\subset\mathbb{R}^n$ with smooth boundary, specifying the pullback of $\alpha$ to $\partial D$, say, $(\partial D)^*\alpha = \phi$, clearly won't work unless $\phi\in \Omega^1(\partial D)$ satisfies $d\phi=0$ and $[\phi]\in H^1_{dR}(\partial D)$ lies in the image of the restriction mapping $H^1_{dR}(D)\to H^1_{dR}(\partial D)$. On the other hand, for the second equation $\Delta\alpha=0$, the appropriate boundary problem would be to specify the restriction (not the pullback) of $\alpha$ to $\partial D$.

Similar remarks apply for $k$-forms with $k>1$ on a general Riemannian manifold with boundary. This problem is not immediately reducible to the Hodge theorem for compact orientable manifolds without boundary.