One of the most cited algorithm in nonlinear manifold learning, with Isomap, is LLE. Contrary to Isomap, LLE tries to retain the local data structure of the sampled manifold. Whereas Isomap preserves absolute distances, LLE preserves local relative distances (it preserves barycenter weights).
This means that LLE is not suitable for every dimensionality reductions. For visualization purposes, it can lead to very different solutions if the manifold is noisy.