"Begründung" = "reasoning" - a line of arguments linked together logically (hopefully)
"Grund" = "motive"
"Ursache" = "cause"
However, there is a confusion about "Grund". Sometimes it is used for "cause" too. Well, a motive is a cause... but not the other way round.
"Was war die Begründung für die Einführung dieser Steuer?" - "What was the reasoning behind the introduction of this tax?"
"Aus welchem Grund hat er das Studium aufgegeben?" - "For what motive has he given up his studies?"
"Wir kennen die Ursache dieses Unfalls nicht." - "We don't know the cause of this accident."
It would be very strange to say "Was war der Grund für diesen Unfall?" This would imply that the accident was not accidental... i.e. that somebody had a motive.
I think Grund more often means "reason." For "motive," I would use the cognate Motiv.
Aus welchem Grund hat er das Studium aufgegeben? - "For what reason has he given up his studies?" Or more simply Warum hat er... - "Why has he..."
For an accident you definitely want to know the cause Ursache, not the reason Grund or motive Motiv.