I wouldn't exactly call it a cheap rip off. It's more like an expensive rip off. For someone not into any of the series they could just as well be interchangeable, with many similar concepts and themes. At the same time there were many differences between the franchises, and a distinct difference in time.
The first Pokemon media, Pokemon Red and Blue, were released in 1996, meaning that when the first Digimon media, a Tomogachi, came out in 1999 Pokemon was already a household name and anchored more firmly in the publics eye than a glue-covered knife.
And that was Pokemon's first main advantage, it came first. Due to the similarities between the two it was inevitable that the latter would be compared to the former. The second main advantage Pokemon had was its primary medium.
Pokemon was and still is incredibly popular in everything from toys to games. The franchise is a gold mine which always sell. Digimon, on the other hand, couldn't keep up the quality or quantity of the Pokemon tsunami. Bar the TV show (which was liked by critics and consumers alike), Digimon simply didn't have the same appeal one their other products which were usually cheap rip offs. So in a sense Digimon was a uneven rip off.
However, that is not what spelled the end of Digimon, but the quality of the show. Most can agree that the Pokemon show isn't that great. It's endless repetition with the same characters year after year after year. Great for entertaining your kids and producing cheap and selling a ton of peripherals for, even though the show really isn't that good.
The Digimon show at the other hand, was really damn good. Albeit clunky in the animation department the show had a surprising depth, in character, tone and story. It had archs, interesting characters with human emotions and reactions and good interactions between the characters. It was overall a great show and far better than the Pokemon show. But the issue with greatness is that it must inevitably end or stay over its welcome. And the creators choose the former, ending the series on a high note. Sure they made sequel series (which I haven't seen past the second one, so I can't speak of their appeal) but none managed to capture audiences the same way again. And since the rest of the Digimon franchise was either based on the first show or not strong enough to hold itself afloat the series fell into obscurity.
Had things been different maybe we wouldn't have seen Digimon as a rip off, but as a worthy rival to Pokemon. The name certainly didn't help, nor did the marketing that made it look like Pokemon 2.0. But it is important to note that Digimon is by no means dead. The series have been running more or less continuously since its inception in 2000, with a new sequel series to the original starting this year. Will this revive the Digimon franchise and finally present us with a worthy contender to Pokemon? As it is just the show I don't think it will even scratch the surface of the Pokemon mountain but it might at least make things interesting for Ash an friends.
So in short, the Digimon show was anything but a rip off, at the same time as the rest of the franchise tried a little to hard to compete with Pokemon, something it couldn't do in the in the long run.
(As for the alleged feud, there have never been any official one. No legal actions or anything have occurred between the two. At the same time fans of both mediums fight over which is best, so that could be called an unofficial feud)