The following doesn't work in maxima but does work in Mathematica (tested through Wolfram Alpha).
sum(binomial(2*n - i, i)*binomial(2*n - 2*i, n-i), i, 0, n)
The solution is, according to WFA,
Binomial[2 n, n] Hypergeometric2F1[-n, -n, -2 n, -1]
Maxima can not simplify the example. I think this is not a bug, but a missing feature.
Moving this bug report to the tracker "Feature requests" and adjusting the title.