There is an excellent explanation of this in Chapter 4 of L. N. Trefethen's Approximation Theory and Approximation Practice (henceforth ATAP; the first 6 chapters are available for free online). I will try to summarize it in your notation, but I recommend reading his explanation.
Chebyshev interpolation
Interpolation at the Chebyshev points gives your $\Pi^{CG}_n f$, which can also be written in terms of the first $n$ Chebyshev polynomials:
$$\Pi^{CG}_n f = \sum_{k=0}^n c_k T_k(x).$$
The question then becomes:
How are the coefficients $c_k$ of the degree-$n$ Chebyshev interpolant related to the coefficients $\hat{f}_k$ of the infinite Chebyshev series?
Aliasing
It turns out that on the Chebyshev grid with $n+1$ points, the following Chebyshev polynomials all take exactly the same values:
$$T_{2jn\pm m}(x), \ \ \ \ j = 0, 1, 2 \dots$$
The above statement is valid as long as $0\le m \le n$ and where the subscript is positive (of course). See Thm. 4.1 of ATAP. Thus each $T_k$ for $k > n$ is "aliased" to appear like some corresponding $T_k$ with $k\le n$.
Relation between the Chebyshev interpolant and the Chebyshev series
Once you know about aliasing, the connection between the coefficients $c_k$ and $\hat{f}_k$ is immediate. Since the Chebyshev interpolant and the infinite Chebyshev series must be equal at the Chebyshev points, we have
$$
\sum_{k=0}^n c_k T_k(x_j) = \sum_{k=0}^\infty \hat{f}_k T_k(x_j),
$$
where $x_j$ is any of the Chebyshev points.
But from the aliasing result, we know that each term with $k>n$ in the right-hand-side sum can be rewritten in terms of $T_k(x_j)$ for some $k\le n$. The result is Thm. 4.2 of ATAP, which gives the formulae (for the degree-$n$ interpolant coefficients)
\begin{align}
c_0 - f_0 & = \sum_{j=1}^\infty f_{2jn} \\
c_n - f_n & = \sum_{j=1}^\infty f_{(2j+1)n} \\
c_k - f_k & = \sum_{j=1}^\infty (f_{2jn+k} + f_{2jn-k}) & 1\le k \le n-1.
\end{align}
So the difference between the coefficients of the Chebyshev interpolant and the Chebyshev projection is given by certain sums of coefficients of the remainder of the Chebyshev series.
One of the major themes of ATAP is that the Chebyshev interpolant is almost as good as the Chebyshev projection for approximation purposes; see Theorems 7.2, 8.2, and 16.1 therein. The great virtue of the interpolant is that it is much easier to compute.
Best Answer
"Approximation" is a very general term, whose definition I'll defer to a later paragraph. For the time being, I concentrate on interpolation and extrapolation.
The assumption here is that you are given a set of points $(x_k,y_k),\quad k=1\dots n$. Interpolation corresponds to finding a function $f(x)$ (e.g. a polynomial) such that $y_k=f(x_k)$ for all $k$.
Usually, the points $(x_k,y_k)$ are within some interval or region, e.g. $x_1 < x_2 < \dots < x_n$. Extrapolation is essentially evaluating the interpolating function $f(x)$ at a value of $x$ outside the region/interval containing the original points, e.g. some value $w < x_1$, or $z > x_n$. This can be slightly risky, since an interpolating function being well-behaved at a neighborhood of the interpolation points does not at all guarantee that the function will still behave nicely outside the original interval/region (and even then you see things like the Runge phenomenon, where the interpolant wiggles widely in between interpolation points).
Approximation, as I've said, is a very general term. One usually does not speak of interpolants as "approximants", since interpolants pass through each given point by construction. However, by a slight abuse of terminology, one sometimes considers the evaluation of some "difficult" function $f(x)$ by taking its values at known points, and constructing an interpolant from these known points. In this respect, the "interpolant" here is called an "approximant". (In my opinion, this (usually) isn't the best way to obtain an approximant!)
One might also consider the more general case of the given points having some form of error; here, one usually fits instead of interpolating. I talked about that in this answer, so I won't be repeating myself...