Your matrix X has full rank, so the difference between the two methods shouldn't be so high. Running it on my pc I indeed obtain:
X = [13.5, 6.75, 6.75;6.75,6.75,0;6.75,0,0]
y = [0.62;0.31;0.31];
format long
a1=X\y
a2=pinv(X)*y
a1 =
0.045925925925926
0
0
a2 =
0.045925925925926
0.000000000000000
-0.000000000000000
They are the same. Differences among the two methods show up when the matrix X has not full rank. Then, if y does not belong to R(X), the range space of X, then pinv method returns the orthogonal projection of y onto R(X). While other methods triggered by \ operator return different approximations, depending on which one is triggered. The collection of methods that \ can trigger (depending on properties of matrix X) is listed here (bottom of the page):
