E5 is an embedding model family trained on large collections of weakly supervised text pairs, with the task described in the input so one model serves queries, passages and classification. It solves the problem that an embedding model trained on one kind of pair transfers poorly to another. It exists because describing the task in the text is a cheap way to make one model cover several retrieval settings.
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.