I have seen these two terms being used interchangeably. For example:
Manifold Learning (often also referred to as non-linear dimensionality reduction) pursuits the goal to embed data that originally lies in a high dimensional space in a lower dimensional space, while preserving characteristic properties.
In this tutorial ‘manifold learning’ and ‘dimensionality reduction’ are used interchangeably.
Dimensionality reduction methods are a class of algorithms that use mathematically defined manifolds for statistical sampling of multidimensional classes to generate a discrimination rule of guaranteed statistical accuracy.
However, http://scikit-learn.org/stable/modules/manifold.html is more nuanced:
Manifold learning is an approach to non-linear dimensionality reduction.
One first difference I can see is that a manifold can be linear, therefore one should compare non-linear manifold learning and non-linear dimensionality reduction.