Example of $c^Tx' = c^Tx$ where x is the optimal solution for the linear relaxation (LP) of x' (ILP)
Consider any problem in which there are several different solutions, for example MAX-CLIQUE on a graph in which there are several cliques of maximum size. Any convex combination of optimal solutions in another optimal solution which isn't integral.

If the optimum value of the integer program is the same as its linear programming relaxation, then any optimal solution to the integer program is also an optimal solution of the linear program. The problem is that there could be other optimal solutions to the linear program, and an LP solver might find one of them instead of an integral optimal solution. It would be, however, inaccurate to say that the LP and ILP have the same value but not the same solution. Rather, all optimal solutions of the ILP are also optimal solutions of the LP. It's just that the LP has more optimal solutions.