The first picture can be found in John Roberts' book on Brighton Hove & District published 1984, showing police plate 56 of Brighton Hove & Preston United, allocated to Castle Square to Preston Park, as can be noted by the side running board. A timetable for this route as a guide to date refers to November 1887. As an amalgamation of horse bus services occured in 1884.  So that's the best guesses I can make from written references to hand.

For the motor bus, this looks like one of the Milnes-Daimler models introduced in 1904, where BH&PU had a fleet of 25 of these by 1907. Horse bus services were changed over to these motor buses by July 1907, with the remaining services in December 1913. By 1916 BH&PU had been taken over by Thomas Tilling. The Portslade to Kemp Town services as noted on the front running board of this bus were a 15 minute service.

The horse bus would seem to date from between 1883, when the Brighton Hove and Preston United Omnibus Company was formed, and the early years of the 20th Century. I can't identify exactly where it was taken, but suspect it might have been near the borough boundary on the London Road, possibly on the corner of what became Knoyle Road. St John's Church would not then have been built.
The motorbus, CD 486, was one of the first half dozen or so acquired by the BH&PUOC, and dated from around 1905. The first bus was bought in 1903, CD 103, and like this one was a Milnes-Daimler. It looks as if the photo was taken at the Rock Street terminus of route 1, about to set off for Portslade.

Thank you gentlemen for your comments regarding these photos. I had no idea that the horse bus photo had been published in a book. I wonder where the author obtained the original from? I should have mentioned that my grandfather's name was Arthur Edwin Pattenden. He was born in 1871, so I imagine that would date the horse bus photo as sometime in the 1890s, judging by his apparent age in the photo. Anyway, interesting history notes by you two experts! Where is (or was) Rock Street?

On the second photo, the bus appears to be on the north side of the road so to your right would be the end of Chesham Road leading to Rock Street, with a public house on both corners, just as the road turns up North and left to Eastern Road. There is a more familiar view of this location with another bus outside the pub sporting an advert for Banfields Ironmongers, still surviving today (2008). As for the original view seen in a book, many enthusiasts or historians collect photographs almost to an obsession but they become very useful when writing a themed book.