Fourier-Legendre series vs Lagrange interpolating polynomial Let's say we want to approximate a regular function f(x) within closed interval [-1,1],
Using Lagrange interpolating series we can write: f(x) = Lj(x)*fj
http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html
Using Fourier-Legendre series we can write: f(x) = aj*Pj(x)
http://mathworld.wolfram.com/Fourier-LegendreSeries.html
Question:
1) What are the main advantages of using Fourier-Legendre series over the Lagrange interpolating series?
2) And disadvantages?
This is what I am guessing:
1) The degree of smoothness is embedded in the Fourier-Legendre interpolant. Meaning we can vary its smoothness at ease without the need of increasing sample points.
2) And vice versa for the Lagrange interpolating series.
3) The approximation using the Lagrange interpolating series is exact at sampling point while as Fouries-Legendre isn't.
This leads to my final question 3 and 4
Q3: What if we have infinite sampling points?
Q4: What if we can afford to have infinite Fouries-Legendre series?
 A: "Warning English isn't my first language"
It might be a bit late, but..
Usually everything that has to do with Fourier works under the assumptions that the function you want to interpolate is periodic and unless you work with signal processing or vibrational mechanics this is hardly never the case, so  this is a big disadvantages of the Fourier interpolation. Where the Lagrange method can be use on every type of function.
you are right about the degree of smoothness depending on the number of points versus the degree of the polynomials used. As the  Lagrange interpolation only increase in smoothness as the number of points increase where you in the Fouirer interpolation mainly depends on the order of polynomials instead. 
About your last questions as they both go to infinity they should both become equal to the function you interpolate perfectly. This is basically the definition of interpolation, so here there should be no difference between the two. What you can ask on the other hand is how fast they converge towards the solution, as one might be faster. On this question however I have to leave you hanging as it properly depends on the type of function you want to interpolate and which points you use. As the interpolation accuracy not only depends on the number of points and the degree of the polynomials used but also on where you choose your points to be. For an example look up "Runge's phenomenon", but in general you'll want many points where the gradient of the functions changes a lot, and few point where the gradient is steady.
as a conclusion I would imagine that the Fourier interpolation is superior when working with periodic functions and the Lagrange interpolations is better on everything else. But as I'm not and expert in this particular field, this is only  speculations, and there might be additional differences between the two.
