partial is used to freeze arguments and keywords. It creates a new callable object with partial application of the given arguments and keywords.

from functools import partial
from operator import add

# add(x,y) normally takes two argument, so here, we freeze one argument and create a partial function.
adding = partial(add, 4)
adding(10)        # outcome will be add(4,10) where `4`  is the freezed arguments.

This is useful when you want to map a list of numbers to a function but maintaining one argument frozen.

# [adding(4,3), adding(4,2), adding(4,5), adding(4,7)]
add_list = list(map(adding, [3,2,5,7])) 

partialmethod was introduced in python 3.4 and it is meant to be used in a class as a method definition rather than been directly callable

from functools import partialmethod
class Live:
    def __init__(self):
        self._live = False
    def set_live(self,state:'bool'):
        self._live = state
    def __get_live(self):
        return self._live
    def __call__(self):
        # enable this to be called when the object is made callable.
        return self.__get_live()

    # partial methods. Freezes the method `set_live` and `set_dead`
    # with the specific arguments
    set_alive = partialmethod(set_live, True)
    set_dead = partialmethod(set_live, False)

live = Live() # create object
print(live()) # make the object callable. It calls `__call__` under the hood
live.set_alive() # Call the partial method
print(live())

As @HaiVu said in his comment partial called in a class definition will create a staticmethod, while partialmethod will create a new bound method which when called will be passed self as the first argument.

I wasn't sure whether the c.get_partialmethod()() call below would work or not, but as z33k mentioned in the comments, it doesn't:

import functools

class Cell:
    def __init__(self):
        pass

    def foo(self, x):
        print(x)

    def get_partial(self):
        return functools.partial(self.foo, True)

    def get_partialmethod(self):
        return functools.partialmethod(self.foo, True)

c = Cell()

print(c.get_partial()) # functools.partial(<bound method Cell.foo of <__main__.Cell object at 0x000001F3BF853AC0>>, True)
print(c.get_partialmethod()) # functools.partialmethod(<bound method Cell.foo of <__main__.Cell object at 0x000001F3BF853AC0>>, True, )

c.get_partial()() # Prints True
c.get_partialmethod()() # TypeError: 'partialmethod' object is not callable