While they work quite differently in terms of implementation, the end result for unsupervised learning is quite similar:
Dynamic time warping measures the distance between timeseries-like data (which can even be images with some clever hacks), given some warping constraints.
Autoencoders based on LSTMs learn representations of timeseries-like data, given some layer constraints (such as bottleneck size, or extensions like variational autoencoders). These representations lie in a space where their mutual distances corresponds to their similarity (at least for the variational kind).
But I haven't seen any direct comparisons for classification/clustering. The most cited pitfall of dynamic time warping seems to be quadratic complexity, which can be reduced to approximately linear with the right constraints. So even in terms of speed there doesn't seem to be a huge difference.
What would be a strong argument for using one over the other?