NNs are typically continuous/differentiable so you can do gradient-based learning on them. We often want to use some of the structure the NN has learned to represent data efficiently. E.g., we might take a pre-trained GPT-type model, and put a passage of text through it, and instead of getting the next-token prediction probability (which GPT was trained on), we just get a snapshot of some of the activations at some intermediate layer of the network. The idea is that these activations will encode semantically useful information about the input text. Then we might e.g. store a bunch of these activations and use them to do semantic search/lookup to find similar passages of text, or whatever.

Quantized embeddings are just that, but you introduce some discrete structure into the NN, such that the representations there are not continuous. A typical way to do this these days is to learn a codebook VQ-VAE style. Basically, we take some intermediate continuous representation learned in the normal way, and replace it in the forward pass with the nearest "quantized" code from our codebook. It biases the learning since we can't differentiate through it, and we just pretend like we didn't take the quantization step, but it seems to work well. There's a lot more that can be said about why one might want to do this, the value of discrete vs continuous representations, efficiency, modularity, etc...