I loved both series and read them all. I get annoyed with people saying O'Brian is better. O'Brian is DIFFERENT. Much of the "action" in Forrester's stories (not just Hornblower, most of his stuff is like this) is inside the character's heads. O'Brian didn't do much stream of consciousness stuff. Usually you see everything from the outside. Forrester mostly keeps his eye on the main character, almost a 1st person kind of feel, though both wrote in the 3rd person. O'B skips around a lot more. O'Brian is frequently funny. Forrester rarely is. Forrester is more tense. In terms of painting a picture of the times, I think Forrester is much more realistic. O'Brian deals with a much wider slice of life though. Hornblower is an idealist, an intellectual, and a naval officer. Aubrey is all that, though his idealism is less thought out and his intellect is more limited in scope, but he is also a landowner, a politician, a lord of the manor, an amateur astronomer, a womanizer. He has at least a nodding acquaintance with 2 of the leading scientists of the day. And Maturin is Doc Savage and 007, with a past. Forrester is careful to keep Hornblower's chronology consistent and believable. O'B didn't try to.
I'm not sure what got O'B started. Forrester started thinking about Hornblower while reading antique bound volumes of The Naval Chronicle of that era.