As of the master branch today (1.15.0), np.product just uses np.prod, and may be deprecated eventually. See MAINT: Remove duplicate implementation for aliased functions. #10653.
And np.prod and ndarray.prod both end up calling umath.multiply.reduce, so there is really no difference between them, besides the fact that the free functions can accept array-like types (like Python lists) in addition to NumPy arrays.
Prior to this, like in NumPy 1.14.2, the documentation claimed np.product and np.prod were the same, but there were bugs because of the duplicated implementation that Parag mentions. i.e. Eric Weiser's example from #10651:
>>> class CanProd(object):
def prod(self, axis, dtype, out): return "prod"
>>> np.product(CanProd())
<__main__.CanProd object at 0x0000023BAF7B29E8>
>>> np.prod(CanProd())
'prod'
So in short, now they're the same, and favor np.prod over np.product since the latter is an alias that may be deprecated.