I don't get "rival" vibes at all. It's definitely a father/difficult teenager dynamic, which helped picturing Anakin's dark nature. You get the friendly, playful tones at the beginning of Episode III. For me that's enough to understand that between TPM and RotS they went through a lot, can get on each other's nerves, but there's still a very strong father/son (or brotherly) bond between them. In addition, there's also The Clone Wars where their relation is developed - it's not a movie, but it's a part of the story and you can't exclude that from the picture completely.