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.
Instructor embeds text together with an instruction describing what the embedding is for, so one model serves retrieval, clustering and classification without fine-tuning for each. It solves the problem that a single fixed embedding cannot be optimal for several different tasks, and fine-tuning a copy per task is expensive and easy to get wrong. It exists because telling the model what kind of similarity you want is a cheap substitute for training it to want that.