The morphism from Bott-Tu cohomolgy to Godbillon cohomology can be made explicit: $$\Psi:H^*(M,N)_{BT}\to H^*(M,N)_G$$
$$(\omega,\theta)\mapsto \omega-d(\pi^*\theta\wedge \eta)\tag{1}\label{1}$$
Here we take a tubular neighborhood $T$ of $N$ in $M$, and let $\pi:T\to N$ be the projection. $\eta$ is a bump function defined in the following way: Take a sequence of tubular neighborhoods $N\subseteq T’’\subseteq T’\subseteq T$ of $N$, such that the closure of the previous one is contained in the next. Then $\eta$ is a smooth function which is constant $1$ in $T’’$ and is zero on $M\setminus T’$.
First, $\Psi$ is well-defined: If $(\omega,\theta)$ is a closed form, then $i^*\omega=d\theta$, so the expression $(\ref{1})$ vanishes on $N$ as a result of $\eta$ being constant $1$ near $N$. Also, one can easily verify that $\Psi$ sends closed forms to closed forms. When $(\omega,\theta)=d(\omega',\theta')$ is exact, we have $$\Psi d(\omega',\theta')=\Psi(d\omega',i^*\omega'-d\theta')=d\omega'-d(\pi^*(i^*\omega'-d\theta')\wedge \eta).$$ One should note that $\omega'-\pi^*i^*\omega'\wedge\eta$ vanishes on $N$, and $d\theta'\wedge\eta=d(\theta'\wedge\eta)\pm\theta'\wedge d\eta$. Since $d\eta$ vanishes on $N$ and $dd(\theta'\wedge\eta)=0$, we conclude $\Psi d(\omega',\theta')=d\lambda$, where $\lambda$ is a differential form on $X$ that vanishes on $N$, so $\Psi$ send exact forms to exact forms. (However, one should note that $\Psi$ is not well-defined on the level of non-closed forms in general.)
Second, one note that $\Psi$ is independent of choice of tubular neighborhoods and the bump function $\eta$, since any of such two will lead to an expression in $(\ref{1})$ differing by an exact form.
Finally, let’s show $\Psi$ is inverse to the map
$$\Theta: H^*(M,N)_{G}\to H^*(M,N)_{BT}$$
$$\alpha\mapsto (\alpha,0).$$
Obviously, $\Psi\circ\Theta=Id$. On the other side,
$$\Theta\circ\Psi(\omega,\theta)=\big (\omega-d(\pi^*\theta\wedge \eta),0\big ),$$
which differ to $(\omega,\theta)$ by an exact form $d(\pi^*\theta\wedge \eta,0)$, so $\Theta\circ\Psi=Id$ on the level of cohomology.
Therefore, the two relative cohomolgy theories are isomorphic.