IIUC an open cover of a set X is defined as a set of open sets which contain the set X. The closure is the union of the set and the limit points. It seems that a closed cover of a set, i.e. a set of closed sets which contain the set X and the closure are similar sets or the same set.
Do you agree?
For example: The set (0,1) has closed cover [0,1] which is the same as the closure.
Please apologize if I got it wrong because the concepts are new to me.
The closure of a set $A$ in topology is defined as the intersection of closed sets containing $A$. Basically, in topology, intersection of closed sets is closed. Since intersection of two sets is smaller than the two sets, one can see that the closure of a set $A$ is the "smallest closed set containing $A$".
However, not every closed set containing $A$ is its closure. For example, the closure of $(0,1)$ in $\mathbb{R}$ is $[0,1]$. $[-1,2]$ is also a closed set containing $A$ but it's not its closure.
Also, note that opennness and closedness are relative properties in topology. The set $(0,1]$ is open in the topology induced on $[0,1]$ from $\mathbb{R}$, but it is not open in $\mathbb{R}$. Similarly, $(0,1]$ is closed in $(0,2)$ in the topology induced on $(0,2)$ in $\mathbb{R}$, but it is not closed in $\mathbb{R}$. So, it's very important to remember that when you talk about openness or closedness, you should always pay attention to your ambient space. Plus, the closure of a set also depends on the ambient space. For example, the closure of $(0,1)$ in $(0,2)$ is $(0,1]$ which is not open in $\mathbb{R}$.
Read about the subspace topology on Wikipedia to understand this stuff better. Also, read theorems $2.24$, $2.27$ and $2.30$ in Rudin's mathematical analysis.