I would tentatively say that there is no better synonym for in other words than the expression to put it another way. This is how the Macmillan Dictionary defines it:

used when you are going to explain something in a different way in order to make it easier to understand

It, of course, means exactly the same thing as put differently. The only difference is that, in terms of the flow, to put it another way sounds a lot smoother than put differently in your particular example. I'd also add that it's just better idiomatic English in general.