Catching a bus describes the process of getting to a stop/station, waiting and boarding.
Taking a bus describes the entire process, including the journey itself.
Much of the time, the distinction isn't important. For example:
"How did you get to work today?"
"I caught a bus." (The listener infers that having caught the bus, you stay on it)
"I took the bus." (You have described the journey)
However, it could be relevant:
"I read a book while I was taking the bus" (yes: reading while the bus is moving)
"I read a book while I was catching the bus" (unlikely: reading while stepping onto the bus, paying the driver, etc.)
It is a similar meaning to catch as catching a fish, or catching a ball. You and the bus are apart, and then you bring yourself into contact with it through your own effort. If you try to catch a bus, and fail, you miss it.
One other thing, you catch a bus at a specific place:
"You can catch the bus to Coventry on Smith Street"
But you take the bus from a place:
"You can take the bus to Coventry from Smith Street"
