Does the k-clique problem became easier on sparse graphs?
Let me not completely agree with Yuval's answer.

Indeed, in some sense sparsity does make a maximum clique problem easier, and this is proved also by solving it on real-life networks. Usually benchmarks for a maximum clique are pretty dense. And here is why.


If it is known an integer $k$ that is a lower bound of the clique number (or the number itself, but we are looking for a clique), then one can reduce the graph to $(k-1)$-core, that is the graph where each node has a degree at least $k-1$. The resulting graph might be considerably smaller in size and thus, for example, IP solver might get a huge boost. How to reduce: delete all vertices with degree smaller than $k-1$ until those exist.
If you look at sparsity not from the density perspective, but from the degeneracy perspective, things are even more formalized. Usually we say that graph is "sparse" when its degeneracy is low. And there exists FPT algorithms with running time around $O^{*}(2^{d/4})$ where $d$ is graph's degeneracy (here big-O star means that the algorithm depends polynomially on the graph size).


Conclusively, the maximum clique problem is immensely easier on sparse instances, though it still remains $\mathcal{NP}$-Complete (but FPT on degeneracy).