Is this the case, or am I missing some fundamental difference?
Though choices of Similarity measure, Laplacian, and Clustering algorithm etc may differ in specific implementations, the general algorithm for Spectral Clustering is indeed identical to that of Laplacian Eigenmaps + a Clustering method:
4.6 Relation between LE and Spectral Clustering
Although Laplacian Eigenmaps are oriented to dimensionality reduction algorithm and Spectral Clustering is focused on cluster data, they are very related.
Both methods are approaches provided by the eigenvectors of Laplacian Graphs. Specifically, if we want to compare them, we must focus our attention on Normalized Spectral Clustering based on Random Walks (algorithm 5). If we compare both algorithms (see comparative table 4.6.1), we will see that the only difference is that Spectral Clustering makes one step more to cluster the information in the new dimension.
- Advanced methods for dimensionality reduction and clustering: Laplacian Eigenmaps and Spectral Clustering, pp.41-42 (2010)
As one can see, steps 1 - 8 are identical for both algorithms, the only difference being whether the embedding itself is returned (LE) or a clustering algorithm is applied and the clusters returned (SC).