np.max is just an alias for np.amax. This function only works on a single input array and finds the value of maximum element in that entire array (returning a scalar). Alternatively, it takes an axis argument and will find the maximum value along an axis of the input array (returning a new array).

>>> a = np.array([[0, 1, 6],
                  [2, 4, 1]])
>>> np.max(a)
6
>>> np.max(a, axis=0) # max of each column
array([2, 4, 6])

The default behaviour of np.maximum is to take two arrays and compute their element-wise maximum. Here, 'compatible' means that one array can be broadcast to the other. For example:

>>> b = np.array([3, 6, 1])
>>> c = np.array([4, 2, 9])
>>> np.maximum(b, c)
array([4, 6, 9])

But np.maximum is also a universal function which means that it has other features and methods which come in useful when working with multidimensional arrays. For example you can compute the cumulative maximum over an array (or a particular axis of the array):

>>> d = np.array([2, 0, 3, -4, -2, 7, 9])
>>> np.maximum.accumulate(d)
array([2, 2, 3, 3, 3, 7, 9])

This is not possible with np.max.

You can make np.maximum imitate np.max to a certain extent when using np.maximum.reduce:

>>> np.maximum.reduce(d)
9
>>> np.max(d)
9

Basic testing suggests the two approaches are comparable in performance; and they should be, as np.max() actually calls np.maximum.reduce to do the computation.

Well from my timings it follows if you already have numpy array a you should use a.max (the source tells it's the same as np.max if a.max available). But if you have built-in list then most of the time takes converting it into np.ndarray => that's why max is better in your timings.

In essense: if np.ndarray then a.max, if list and no need for all the machinery of np.ndarray then standard max.

They may differ in edge cases, such as a list containing NaNs.

import numpy as np
a = max([2, 4, np.nan])     # 4
b = np.max([2, 4, np.nan])  # nan

NumPy propagates NaN in such cases, while the behavior of Python's max is less certain.

There are also subtle issues regarding data types:

a = max([10**n for n in range(20)])     # a is an integer
b = np.max([10**n for n in range(20)])  # b is a float

And of course running time differences documented in numpy.max or max ? Which one is faster?

Generally, one should use max for Python lists and np.max for NumPy arrays to minimize the number of surprises. For instance, my second example is not really about np.max but about the data type conversion: to use np.max the list is first converted to a NumPy array, but elements like 10**19 are too large to be represented by NumPy integer types so they become floats.