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.
GTE is a family of text embedding models trained for retrieval, with variants covering English and multilingual use and long inputs. It solves the same problem as the other retrieval-focused embedding families: a general-purpose encoder is not optimised for the asymmetric comparison of a short query against a long document. It exists because retrieval quality is decided at the embedding step, and a family that scores well on public retrieval benchmarks is the cheapest way to improve everything downstream.