(written from a Production point of view)
Breezy was the beagle that worked as back-up and then took over the role of Porthos after the episode "Two Days and Two Nights". Breezy was a bit more "spunky" than Prada, so, when they needed a more playful dog during the first season, Breezy was used – most famously, in the "where no dog has gone before" shuttlepod exit on location outdoors in "Strange New World". After the first season, Breezy took over the role full-time.
Although Porthos was a male dog, as was Prada, the series' producers didn't want the audience's attention drawn to Prada's "male components", because the idea was to replace him with Breezy, a female, and hope the audience wouldn't notice the change. They also had to paint a temporary black spot on Breezy's side during shooting in order to match Prada's color.