A chain map $\Theta$ from the Godbillon theory to the Bott-Tu version is given by $\omega \mapsto (\omega,0)$ (note that is a chain map only on $\Omega^{p} (M;N)_{G}$).
I claim that this induces an isomorphism on cohomology. A couple of special cases is obvious: if $N=\emptyset$, then both theories agree with absolute de Rham theory.
If $N \to M$ is a homotopy equivalence, both theories are trivial by long exact sequences and homotopy invariance of the absolute theory.
For the general case, pick a tubular neighborhood $U$ of $N$. You get short exact sequences of chain complexes (in both cases)
$$
0\to \Omega (M;N) \to \Omega(U;N) \oplus \Omega (M-N) \to \Omega (U-N) \to 0
$$
(exactness is checked by means of a partition of unity), and $\Theta$ compares the both short exact sequences. The associated (Mayer-Vietoris) exact sequence and the $5$-lemma concludes the proof.
Suppose that $C$ is a closed subset of $M$. Denote by $\newcommand{\eO}{\mathscr{O}}$ $\eO$ its complement.
The DeRham cohomology of $M$ is in fact the cohomology associated to a particular soft resolution of the constant sheaf $\newcommand{\ur}{\underline{\mathbb{R}}}$ $\ur$ on $M$.
To any sheaf $\newcommand{\eS}{\mathscr{S}}$ $\eS$ on $M$ we can associate a sheaf $\eS_{\eO}$, also on $M$ whose stalk $\eS_{\eO}(x)$ at $x\in M$ is $\eS(x)$ if $x\in \eO$, and $0$ if $x\in M\setminus \eO$. For any open subset $U\subset M$ the space $\Gamma(U, \eS_{\eO})$ of sections of $\eS_{\eO}$ over $U$ consists of the section $s\in \Gamma(\eO\cap U,\eS)$ such that the support of $s$ is closed in $U$. The operation $\eS\mapsto \eS_{\eO}$ is an exact functor. Moreover if $\eS$ is soft, so is $\eS_{\eO}$. Thus the DeRham resolution
$$ 0\to\ur\to\Omega^0\to\Omega^1\to\cdots, $$
$\Omega^k=$ the sheaf of smooth $k$-forms on $M$, produces a soft resolution of $\ur_{\eO}$
$$ 0\to\ur_{\eO}\to\Omega_{\eO}^0\to\Omega_{\eO}^1\to\cdots . $$
Hence, the cohomology of the sheaf $\ur_{\eO}$ is computed by the the cohomology of the complex
$$ \Gamma(M, \Omega_{\eO}^0)\stackrel{d}{\to}\Gamma(M, \Omega^1_{\eO})\stackrel{d}{\to}\cdots, \tag{1} $$
where, as explained above $\Gamma(M, \Omega^k_{\eO})$ consists of smooth $k$-forms on $\eO$ whose support is a closed subset in $M$. Equivalently $\Gamma(M, \Omega^k_{\eO})$ consists of forms on $M$ whose supports do not intersect $C$. If $M$ is compact, then $\Gamma(M, \Omega^k_{\eO})$ consists of form with compact support contained in $\eO$.
On the other hand, the cohomology of $\ur_{\eO}$ can be identified with the relative cohomology $H^\bullet(M,C;\mathbb{R})$. The complex (1) is the a DeRham model for this cohomology. For more details I recommend the comprehensive book of Kashiwara and Schapira Sheaves on Manifolds or my notes which are less comprehensive, but may guide you through the literature. In particular, see Remark 2.17 of my notes.
Best Answer
The proof of the existence of good covers is contained in Bott & Tu on pages 42-43, though that proof does refer out to Spivak (see below).
In general, if your main goal is to study (algebraic) topology of manifolds, you probably don't need to know much about metrics and connections and that sort of thing that typical differential geometry books spend a lot of time on. It sounds like what you really need is some material on differential topology. I don't know how much they'll cover of the specific things you're looking for, but here are a few suggestions to check out:
Topology and Geometry by Glen Bredon. This might be a particularly good book for you as it really combines the two topics pretty well.
Michael Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1. Despite the title, the first volume is more about differential topology than geometry. Also this is what Bott and Tu cite for their key fact needed for the existence of good covers.
Differential Topology by Guillemin and Pollack - this is a very readable introduction.
Introduction to Smooth Manifolds by John M. Lee - this is oriented a bit more toward geometry but you can find a lot in it.
I'd recommend skimming through these (and others) to find one that suits you and has the kinds of things you're looking for.