The Neyman-Pearson lemma is a result about constructing MP tests in the simple-null simple-alternative case, whereas LRT's are just forms of tests. Typically the NP-Lemma states that in the simple-null simple-alternative case, an appropriately chosen LRT is most powerful of a desired size (say alpha). However, particularly in the discrete case, a LRT of size alpha may or may not exist whereas the NP-Lemma, by utilizing randomized tests (which are NEVER LRT's), can always specify a most powerful size alpha test.
One implication of the Neyman-Pearson lemma is that in the simple-null simple-alternative setup, a Likelihood Ratio Test is always most powerful of its size. The Neyman-Pearson lemma goes beyond this however, stating that a most powerful size alpha test always exists and states how to construct it, while a size alpha LRT doesn't necessarily exist.
Outside of the simple-null simple-alternative setup, the NP-Lemma does not apply whereas it is always possible to construct a LRT.