BGE is a family of embedding models trained specifically for retrieval, available in sizes small enough to run locally. It solves the problem that a general-purpose embedding model is not optimised for the asymmetric comparison a search query and a document require. It exists because retrieval quality is set at the embedding step, so a model trained for it improves everything downstream without changing anything else.
Nomic Embed is an open embedding model released with an open training dataset and reproducible training run, as well as long-context support. It solves two problems at once: most open embedding models publish weights but not the data they were trained on, which makes their behaviour hard to reason about, and most cap inputs short of a document. It exists because an embedding model whose training data is inspectable can be evaluated for the biases that come from that data.