The phrase "break out" is somewhat more figurative and puts the focus on the general idea of bringing something out, rather than the physical act. Upon receiving good news, one might want to "break out the champagne", connoting the idea of celebrating with a drink. In contrast, to "take out the champagne" might be to simply retrieve a bottle from the refrigerator.
Here, "break out the checkbook" and "take out the checkbook" are nearly interchangeable, but I'd say the former more calls attention to the fact that money is being spent, rather than the form of payment. It's a somewhat more emphatic way of referring to the same action, possibly used to highlight that one is excited (or not excited) to make the payment.
There are many cases where either phrase would be appropriate, but here are a couple of examples where the phrases would not be interchangeable:
Go into my purse and take out the checkbook.
I just won the lottery, it's time to break out the checkbook!
The former describes nothing more than the literal physical act of removing the checkbook from the purse - it might not even imply you're going to use it, if you wanted to leave it behind or show it to someone. The latter phrase, on the other hand, figuratively implies that you intend to spend money - it might not even imply that you're literally going to be paying by check.