There is a significant difference in that the convert holders tend to maintain a long-term short position in the stock that they delta-hedge (so the size of their short position goes up and down inversely with stock price moves). With a straight offering, you get immediate flippers, and to some extent, folks that short ahead of the offering, although I believe that is much less prevalent than it used to be. So with a straight offering you don't usually see a prolonged uptick in the short position.
At the end of the day, if the stock price goes up above the conversion price then (as you said) there will be fewer shares outstanding than if they had done a straight share offering. But if the stock price does not appreciate enough, the company will have to find some other way to pay back the debt.