The relation is easier to understand if $G$ is a discrete group. Then your definition is equivalent to an action of $G$ on an object $M$ in $D^b(X)$. This means that for each $g\in G$ we have a morphism $\phi_g\colon M\rightarrow M$ in $D^b(X)$ with, and this is the important part, $\phi_{gh}=\phi_g\phi_h$ in $D^b(X)$. Seen from the point of view of complexes (i.e., before we pass to $D^b(X)$) this means that we have a homotopy (or something slightly worse depending on what kind of complex $M$ is) $\phi_{gh}\sim\phi_g\phi_h$. Experience tells us that such actions up to homotopy is too weak a notion to be useful (things are a little bit tricky as there are non-trivial situations when such an action can in fact be replaced up to homotopy by a true action) and in general one should demand an actual action of $G$ on the complex $M$. The same goes (and in fact even more so) for morphisms, there may be too many morphisms if you just look at morphisms in $D^b(X)$ that commute in $D^b(X)$ with the action of $G$.

Addendum: Donu makes an excellent point and I just want to elaborate. Take an additive $G$-action on some $\mathbb Z/p^n$ which does not lift to an additive action on $(\mathbb Z/p^2)^n$. Then we may look at the morphism $\mathbb Z/p^n \to \mathbb Z/p^n[1]$ whose mapping cone is $(\mathbb Z/p^2)^n$. It will be a $G$-map $D^b(pt)$ but there is no compatible action of $G$ on a mapping cone.

The relation is easier to understand if $G$ is a discrete group. Then your definition is equivalent to an action of $G$ on an object $M$ in $D^b(X)$. This means that for each $g\in G$ we have a morphism $\phi_g\colon M\rightarrow M$ in $D^b(X)$ with, and this is the important part, $\phi_{gh}=\phi_g\phi_h$ in $D^b(X)$. Seen from the point of view of complexes (i.e., before we pass to $D^b(X)$) this means that we have a homotopy (or something slightly worse depending on what kind of complex $M$ is) $\phi_{gh}\sim\phi_g\phi_h$. Experience tells us that such actions up to homotopy is too weak a notion to be useful (things are a little bit tricky as there are non-trivial situations when such an action can in fact be replaced up to homotopy by a true action) and in general one should demand an actual action of $G$ on the complex $M$. The same goes (and in fact even more so) for morphisms, there may be too many morphisms if you just look at morphisms in $D^b(X)$ that commute in $D^b(X)$ with the action of $G$.

Addendum: Donu makes an excellent point and I just want to elaborate. Take an additive $G$-action on some $(\mathbb Z/p)^n$ which does not lift to an additive action on $(\mathbb Z/p^2)^n$. Then we may look at the morphism $(\mathbb Z/p)^n \to (\mathbb Z/p)^n[1]$ whose mapping cone is $(\mathbb Z/p^2)^n$. It will be a $G$-map $D^b(pt)$ but there is no compatible action of $G$ on a mapping cone.

The relation is easier to understand if $G$ is a discrete group. Then your definition is equivalent to an action of $G$ on an object $M$ in $D^b(X)$. This means that for each $g\in G$ we have a morphism $\phi_g\colon M\rightarrow M$ in $D^b(X)$ with, and this is the important part, $\phi_{gh}=\phi_g\phi_h$ in $D^b(X)$. Seen from the point of view of complexes (i.e., before we pass to $D^b(X)$) this means that we have a homotopy (or something slightly worse depending on what kind of complex $M$ is) $\phi_{gh}\sim\phi_g\phi_h$. Experience tells us that such actions up to homotopy is too weak a notion to be useful (things are a little bit tricky as there are non-trivial situations when such an action can in fact be replaced up to homotopy by a true action) and in general one should demand an actual action of $G$ on the complex $M$. The same goes (and in fact even more so) for morphisms, there may be too many morphisms if you just look at morphisms in $D^b(X)$ that commute in $D^b(X)$ with the action of $G$.

The relation is easier to understand if $G$ is a discrete group. Then your definition is equivalent to an action of $G$ on an object $M$ in $D^b(X)$. This means that for each $g\in G$ we have a morphism $\phi_g\colon M\rightarrow M$ in $D^b(X)$ with, and this is the important part, $\phi_{gh}=\phi_g\phi_h$ in $D^b(X)$. Seen from the point of view of complexes (i.e., before we pass to $D^b(X)$) this means that we have a homotopy (or something slightly worse depending on what kind of complex $M$ is) $\phi_{gh}\sim\phi_g\phi_h$. Experience tells us that such actions up to homotopy is too weak a notion to be useful (things are a little bit tricky as there are non-trivial situations when such an action can in fact be replaced up to homotopy by a true action) and in general one should demand an actual action of $G$ on the complex $M$. The same goes (and in fact even more so) for morphisms, there may be too many morphisms if you just look at morphisms in $D^b(X)$ that commute in $D^b(X)$ with the action of $G$.

Addendum: Donu makes an excellent point and I just want to elaborate. Take an additive $G$-action on some $\mathbb Z/p^n$ which does not lift to an additive action on $(\mathbb Z/p^2)^n$. Then we may look at the morphism $\mathbb Z/p^n \to \mathbb Z/p^n[1]$ whose mapping cone is $(\mathbb Z/p^2)^n$. It will be a $G$-map $D^b(pt)$ but there is no compatible action of $G$ on a mapping cone.