Great question. In general, as long as you're making on-time payments, either type of card can benefit your credit score. But there are some important distinctions between credit cards and charge cards, beyond the fact that you generally can't carry a balance on a charge card.
One difference between a credit card and a charge card is that a charge card doesn't generally have a stated limit and therefore it might appear to report as maxed out. Of course the credit bureaus are aware of this and I believe they try to address it, but you know how that goes (i.e. I don't expect they completely correct for it). How it will affect your credit depends a lot on other factors in your credit file - especially how many other cards you have and the utilization on them.
I like charge cards for some who are working on forming good habits around credit cards since you have to pay a charge card in full each month. However, all charge cards have an annual fee. So, I would say, if you don't have a specific reason that makes the card worth the fee (like consistently paying in full, really taking advantage of the rewards and other perks, etc.), then you should probably just go for a credit card - especially if you're just looking for a card to help you improve your already decent credit score.
If you don't pay off a charge card balance, then you can expect a late fee (usually $35), and if you blow it two months in a row, it'll be another $35 or about 3 percent of the balance, whichever is higher. And then, of course, your credit score will take a hit. There's a reason American Express cards are intended for people with good to excellent credit. If you have poor credit or no credit, they may not be a realistic choice at this time. But if you have good credit already, an American Express card can make it better.