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I know that they both represent same polynomial and their formulas, but what is difference between them. If they are not different, then why do we study them separately?

Don't need detailed proof but general idea - 'what is going on?'

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What is great with Newton's interpolation is the fact that if you add new points you don't have to re-calculate all the coefficients (see forward divided difference formula) which can be really useful ! You'll have more details at the end of this article : https://en.wikipedia.org/wiki/Newton_polynomial

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This is the same polynomial but you just find it in different ways. It's always better to have different ways because that way you have a lot more options. For example, if you want to have an easy formula for the remainder of the interpolation then it is much better to work with Newton's method. Another advantage is that if you found the interpolation polynomial in the points $x_0,x_1,...,x_n$ and then you want to add the point $x_{n+1}$ then using Newton's method you can use the polynomial in the points $x_0,...,x_n$ to easily find the polynomial for the points $x_0,...,x_{n+1}$. You don't have to find all the coefficients all over again.

On the other hand, if you want to do Numerical differentiation or Numerical integration then it is much easier to work with Lagrange's method. So having different ways to calculate the interpolation polynomial gives a lot more options. It all depends on what exactly you need to do.

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