Denote the independence number of $G$ by $\alpha(G)$, and the clique cover number by $\overline{\chi}(G)$. It is obvious that $\overline{\chi}(G) \ge \alpha(G)$. You are asking when $\alpha(G) = \overline{\chi}(G)$.

This seems to be the question that drove the definition and investigation of perfect graphs. See Berge's historical overview. The definition of perfect graphs requires the equality between independence number and clique cover number *for every induced subgraph*. This is Berge's "Beautiful Property", defining what he calls "class 2". Rephrasing a comment by Lovász, the induced subgraph condition is needed to remove the artificial examples where a large set $X$ of new vertices is added to a graph such that each vertex in $X$ is adjacent to each vertex of the original graph. Applying this construction to any graph yields a new graph that satisfies your condition.

Berge also defined "class 3", those graphs for which the chromatic number equals the clique number for every induced subgraph, and "class 4", the graphs which contain no induced odd hole or induced odd antihole. In the paper mentioned above, Lovász proved the perfect graph theorem, that the complement of a perfect graph is perfect. This shows that "class 2" and "class 3" are the same. Perfect graphs are now usually defined as "class 3", and the strong perfect graph theorem shows that "class 4", of Berge graphs, is the same as "class 3". Berge claimed he was originally more interested in whether "class 2" and "class 3" coincided.

One final remark: if $\alpha(G) = \overline{\chi}(G)$, then the Shannon capacity of $G$ is $\log \alpha(G)$. This is Berge's "class 1" (note that the "log" is missing in the paper, clearly a typo). This is really saying that in the limit, when $\alpha(G) = \overline{\chi}(G)$ then the independence number dominates the channel performance in the long run. From this applied point of view, it makes sense to look for the kinds of graphs which have this property inherently, and not just by a large independent set being glued on.

However, there may be other applications for graphs for which $\alpha(G) = \overline{\chi}(G)$ holds, but where this relationship fails for some induced subgraphs. This would then make it interesting to find the graphs that have this property but not via a glued-on independent set. To end this answer on a question: do you have such an application in mind?