Why does "delaunay" sometimes give a different result from "delaunayTriangulation"?

I have a file containing a set of x,y,z points which describe the surface of a 10 pence coin. I wish to triangulate the data. I tried Matlab's in-built "delaunay" function and the results are satisfactory. However, if I use Matlab's in-built "delaunayTriangulation" function I get a completely different result. Can you suggest possible reasons for this difference? For visualisation of the problem I have included the script below and I have also attached a .csv file containing a sample of the x,y,z data:

Thanks

    fname = 'ten_pence_coin_sample.csv';
    data = csvread(fname);

    deltaX = 20;   
    TopThreshold = 37000;  
    BotThreshold = 8620;   
    XXXX = 0;
    PrevEnc = 0;
    x = zeros(length(data(:,1)),1);

    for n = 1: length(data(:,1))

        if n>1
           PrevEnc = data(n-1, 4); 
        end  
        Enc = data(n,4);
        if Enc>TopThreshold && PrevEnc < TopThreshold
           XXXX = XXXX + deltaX;   
           x(n) = nan;
        elseif Enc<BotThreshold && PrevEnc > BotThreshold
           XXXX = XXXX + deltaX;      
           x(n) = nan;
        else
           x(n) = XXXX;
        end
    end

    z=data(:,2); 
    z(find(z<0)) = nan; 

    filt_ind = isnan(z) | isnan(x) ;

    z_filt = z ;
    x_filt = x ;
    y_filt = data(:,4) - min(data(:,4)) ;

    z_filt(filt_ind) = [];
    x_filt(filt_ind) = [];
    y_filt(filt_ind) = [];

    DT = delaunay(x_filt,y_filt);
    DT2 = delaunayTriangulation(x_filt,y_filt);

    figure
    trimesh(DT,x_filt,y_filt,z_filt);

    figure
    trimesh(DT2.ConnectivityList,x_filt,y_filt,z_filt);

Answer 1

[John D'Errico]

What is the delaunay triangulation of the unit square? Thus, if we start with the vertices

V = [0 0;0 1;1 0;1 1];

Is there a unique triangulation? I think the answer is clear. There is no unique choice that is better. We might have the pair or triangles

tri1 = [1 2 3;2 3 4]

or we might choose the pair

tri2 = [1 2 4;1 3 4];

Either is as good a choice. Effectively, the difference is only in the diagonal of the square we choose. Two different algorithms might make different choices, and be equally correct in their choice.

In addition, some algorithms resolve issues by making a joggle of the points, a pseudo-random perturbation of points, which can greatly help. However, there you will even find the same algorithm produces non-identical results upon repeated applications.