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.
Cohere Embed is a hosted embedding model that distinguishes the query from the document being indexed, and supports a hundred languages in one model. It solves the asymmetry problem explicitly: a question and a passage are not the same kind of text, and encoding them with the same mode is a compromise. It exists because multilingual retrieval otherwise means either a separate model per language or one model doing all of them badly.