LCA solves a class of problems that come up constantly in tree algorithms. If you want the distance between 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 traversal per query.
LCA also enables path queries. To find the maximum edge weight between nodes, you split the path at the LCA and query each half separately. Version control systems use LCA to find the merge base of branches. In competitive programming, LCA appears in tree DP, path aggregation, and distance problems.