The similarities are vast, given that the commonwealth derived from the empire. But in this case, the commonwealth was a political structure that emerged as the British Empire increasingly devolved power from London to Canada, Australia, New Zealand, etc. This was done gradually in a series of conferences, but the most important was the Balfour conference in 1926.

Eventually, the idea that all of the states where in general, equal took more and more hold (though in reality it was not until the post war period when the UK was stripped of most of her colonies that this was more true then false).

tl;dr - it was the evolution of the British empire, and while similar in culture, radically different in power structure.