Brands do have strengths and weaknesses, but overall there's nothing one can do that the others can't. However sometimes it is easier or there is more choice - eg canon and nikon have the greater range of new own brand and 3rd party lenses for their systems.
I would say at present Nikon is the better choice in terms of their ISO performance which is significantly better than Canons, esp with restoring detail in darker conditions.
However Canon has a powerful range of mid-level longer focal length lenses - 300mm f4, 100-400mm, 400mm f5.6 etc... Which makes them popular with wildife and nature photographers in general.
I would say for your budget a standard entry level camera body and a 70-300mm lens would do you well. At that rough price level the 70-300mm will be decent in quality. Sure there are better lenses (optically speaking) but they will cost and when you are starting out sometimes its better to have a good, but not top rate tool which is more affordable to learn with and so that you can better find out what works for you before spending larger amounts.
For most zoo photography 70-300mm is going to work well for your needs. I have fond memories of mine which I used with a dirt cheap tripod (those aluminium ones that are light as anything and cheaply made, but which work decently well at giving some support).
Some 70-300mm (Eg Sigma 70-300mm APO) have a "macro" mode which is really close up not full macro, but its good enough for butterflies, dragonflies and flowers. Gives a bit more versatility.
I can't recommend a specific 70-300mm as I've not been in the market for one in a long while, but Canon, Sigma, Nikon, Tamron or Tokina would be good solid brands to pick from to suit your budget.