These are known as acyclic spaces (note that since $C_*(X)$ is a bounded below complex of projectives, it being contractible is equivalent to its homology being trivial).
There's an extensive literature about them, starting with this Emanuel Farjoun's paper.
In general, yes $C_*(X)$ does forget some information about $X$: first of all it factors through the stable homotopy type $\Sigma^\infty_+X$ of $X$, and it forgets further information from there (the slogan is that it remembers only the "$\mathbb{Z}$-linear" information contained in $\Sigma^\infty_+X$). For example it is unable to distinguish between $\mathbb{CP}^2$ and $S^2\vee S^4$ (since the attaching map $\eta:S^3\to S^2$ is sent to a map homotopic to 0 by $C_*(-)$).