Explanations of algorithms:
The Neville method is basically polynomial fitting of all
the data points.  Because a polynomial of degree n can have
up to n-1 relative maxima, the phenomenon of "polynomial
wiggle" can have an adverse effect in a region with no data
points, such as the one for this problem.  It is seldom a
good idea to use a polynomial of degree 6 or above unless
it is known that the true function is a polynomial. 
Therefore the Neville method uses successively higher degrees
for the interpolating polynomial.  When the value begins to 
diverge, it becomes noticeable by a human (or a computer) 
looking at the matrix data.  
The accuracy of the Neville procedure is quite good because it
senses when it is diverging (at least the human does).
I do not know about the accuracy of the spline method in matlab,
because it only outputs one number, not a movie of the procedure.
However, knowing matlab, it is probably pretty accurate.
They are both pretty close, and hence both probably pretty good
estimates.