LCA solves a class of problems that come up constantly in tree algorithms. If you want the distance between 2 nodes, you need their LCA. The formula is: dist(u, v) = depth[u] + depth[v] - 2 * depth[lca(u, v)]. Without LCA, computing tree distances requires O(n) traversal per query.
LCA also enables path queries. To find the maximum edge weight between 2 nodes, you split the path at the LCA and query each half separately. Version control systems use LCA to find the merge base of 2 branches. In competitive programming, LCA appears in tree DP, path aggregation, and distance problems.