What are the differences between add_axes and add_subplot?

Common grounds

Both, add_axes and add_subplot add an axes to a figure. They both return a (subclass of a) matplotlib.axes.Axes object.

However, the mechanism which is used to add the axes differs substantially.

add_axes

The calling signature of add_axes is add_axes(rect), where rect is a list [x0, y0, width, height] denoting the lower left point of the new axes in figure coodinates (x0,y0) and its width and height. So the axes is positionned in absolute coordinates on the canvas. E.g.

fig = plt.figure()
ax = fig.add_axes([0,0,1,1])

places a figure in the canvas that is exactly as large as the canvas itself.

add_subplot

The calling signature of add_subplot does not directly provide the option to place the axes at a predefined position. It rather allows to specify where the axes should be situated according to a subplot grid. The usual and easiest way to specify this position is the 3 integer notation,

fig = plt.figure()
ax = fig.add_subplot(231)

In this example a new axes is created at the first position (1) on a grid of 2 rows and 3 columns. To produce only a single axes, add_subplot(111) would be used (First plot on a 1 by 1 subplot grid). (In newer matplotlib versions, add_subplot() without any arguments is possible as well.)

The advantage of this method is that matplotlib takes care of the exact positioning. By default add_subplot(111) would produce an axes positioned at [0.125,0.11,0.775,0.77] or similar, which already leaves enough space around the axes for the title and the (tick)labels. However, this position may also change depending on other elements in the plot, titles set, etc.
It can also be adjusted using pyplot.subplots_adjust(...) or pyplot.tight_layout().

In most cases, add_subplot would be the prefered method to create axes for plots on a canvas. Only in cases where exact positioning matters, add_axes might be useful.

Example

import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = (5,3)

fig = plt.figure()
fig.add_subplot(241)
fig.add_subplot(242)
ax = fig.add_subplot(223)
ax.set_title("subplots")

fig.add_axes([0.77,.3,.2,.6])
ax2 =fig.add_axes([0.67,.5,.2,.3])
fig.add_axes([0.6,.1,.35,.3])
ax2.set_title("random axes")

plt.tight_layout()
plt.show()

Alternative

The easiest way to obtain one or more subplots together with their handles is plt.subplots(). For one axes, use

fig, ax = plt.subplots()

or, if more subplots are needed,

fig, axes = plt.subplots(nrows=3, ncols=4)

The initial question

In the initial question an axes was placed using fig.add_axes([0,0,1,1]), such that it sits tight to the figure boundaries. The disadvantage of this is of course that ticks, ticklabels, axes labels and titles are cut off. Therefore I suggested in one of the comments to the answer to use fig.add_subplot as this will automatically allow for enough space for those elements, and, if this is not enough, can be adjusted using pyplot.subplots_adjust(...) or pyplot.tight_layout().