I don't believe there is any difference between natural sugar and added sugar. Depending on the chemical make-up, I believe even the difference between sugar and HFCS is something like 5% and while HFCS is worse for you, it is not that much worse.
The problem with sugar comes from the amount that people intake every day and how those calories add up and create weight and health problems which puts a financial burden on the whole of society. So, it's not that there is much of a difference between added sugar and natural sugar, it's that when you take into account added sugar upon what sugar is already in that food naturally, you are upping the individual's total sugar intake but because it is added, that can be changed pretty easily.
Also, sugar is not 100% bad, it plays its role. I believe it's very important for a lot of amino acids. It's just that American food has evolved into everything having a shit load of sugar in it because so much is added to it. And all those added sugars add up. I actually believe the WHO suggests an even smaller daily intake compared to the FDA, the WHO says in general (and this depends on your height and weight), women should not take in more that 25g of added sugar a day and that men should not take in more 36g. The FDA's current recommendation is 50g, I believe. Getting below 50g will be tough for most people to do (I imagine most people probably have like 80-100g on a normal day and like 150g-200g at a holiday party with cake and snacks), personally getting below 36g was really tough to do just because of how everything, even things you wouldn't expect, are so loaded with sugar.
I don't actually know what the ideal total sugar intake is (when you add both natural and added together), and I don't think this is a recommendation to suggest people should aim for whatever that number would be, it is more that sugar intake is too high and needs to be reigned in and added sugar is the easier lever to adjust in this situation.
Edit: also, while I again don't think there is that much difference between natural sugar and added sugar, fruit and vegetables and such which have a natural amount of sugar also include a lot of fiber which helps slow down the digestion of sugar so it doesn't hit your liver all at once. But eating like 4 apples at once is not considered healthy, in fact more than 3 servings (whole fruits) a day and not spaced apart is considered too much from what I've read, even with all the fiber you're getting, too much sugar is too much sugar.