TL, DR: There is no difference and they can be used interchangeably.

Besides having the same value as math.inf and float('inf'):

>>> import math
>>> import numpy as np

>>> np.inf == float('inf')
True
>>> np.inf == math.inf
True

It also has the same type:

>>> import numpy as np
>>> type(np.inf)
float
>>> type(np.inf) is type(float('inf'))
float

That's interesting because NumPy also has it's own floating point types:

>>> np.float32(np.inf)
inf
>>> type(np.float32(np.inf))
numpy.float32
>>> np.float32('inf') == np.inf  # nevertheless equal
True

So it has the same value and the same type as math.inf and float('inf') which means it's interchangeable.

Reasons for using np.inf

1. It's less to type:

• np.inf (6 chars)
• math.inf (8 chars; new in python 3.5)
• float('inf') (12 chars)

That means if you already have NumPy imported you can save yourself 6 (or 2) chars per occurrence compared to float('inf') (or math.inf).

2. Because it's easier to remember.

At least for me, it's far easier to remember np.inf than that I need to call float with a string.

Also, NumPy defines some additional aliases for infinity:

    np.Inf
    np.inf
    np.infty
    np.Infinity
    np.PINF

It also defines an alias for negative infinity:

    np.NINF

Similarly for nan:

    np.nan
    np.NaN
    np.NAN

3. Constants are constants

This point is based on CPython and could be completely different in another Python implementation.

A float CPython instance requires 24 Bytes:

    >>> import sys
    >>> sys.getsizeof(np.inf)
    24

If you can re-use the same instance you might save a lot of memory compared to creating lots of new instances. Of course, this point is mute if you create your own inf constant but if you don't then:

    a = [np.inf for _ in range(1000000)]
    b = [float('inf') for _ in range(1000000)]

b would use 24 * 1000000 Bytes (~23 MB) more memory than a.

4. Accessing a constant is faster than creating the variable.

    %timeit np.inf
    37.9 ns ± 0.692 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops each)
    %timeit float('inf')
    232 ns ± 13.9 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
   
    %timeit [np.inf for _ in range(10000)]
    552 µs ± 15.4 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
    %timeit [float('inf') for _ in range(10000)]
    2.59 ms ± 78.7 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
   

Of course, you can create your own constant to counter that point. But why bother if NumPy already did that for you.