What is the difference between np.sum and np.add.reduce?
While the docs are quite explicit:

For example, add.reduce() is equivalent to sum().

The performance of the two seems to be quite different: for relatively small array sizes add.reduce is about twice faster.

$ python -mtimeit -s"import numpy as np; a = np.random.rand(100); summ=np.sum" "summ(a)"
100000 loops, best of 3: 2.11 usec per loop
$ python -mtimeit -s"import numpy as np; a = np.random.rand(100); summ=np.add.reduce" "summ(a)"
1000000 loops, best of 3: 0.81 usec per loop

$ python -mtimeit -s"import numpy as np; a = np.random.rand(1000); summ=np.sum" "summ(a)"
100000 loops, best of 3: 2.78 usec per loop
$ python -mtimeit -s"import numpy as np; a = np.random.rand(1000); summ=np.add.reduce" "summ(a)"
1000000 loops, best of 3: 1.5 usec per loop

For larger array sizes, the difference seems to go away:

$ python -mtimeit -s"import numpy as np; a = np.random.rand(10000); summ=np.sum" "summ(a)"
100000 loops, best of 3: 10.7 usec per loop
$ python -mtimeit -s"import numpy as np; a = np.random.rand(10000); summ=np.add.reduce" "summ(a)"
100000 loops, best of 3: 9.2 usec per loop

Short answer: when the argument is a numpy array, np.sum ultimately calls add.reduce to do the work. The overhead of handling its argument and dispatching to add.reduce is why np.sum is slower.

Longer answer: np.sum is defined in numpy/core/fromnumeric.py. In the definition of np.sum, you'll see that the work is passed on to _methods._sum. That function, in _methods.py, is simply:

def _sum(a, axis=None, dtype=None, out=None, keepdims=False):
    return um.add.reduce(a, axis=axis, dtype=dtype,
                            out=out, keepdims=keepdims)

um is the module where the add ufunc is defined.