Those are different plotting methods. Fundamentally, they both produce a matplotlib object, which can be shown via one of the matplotlib backends.
There is however an important difference. Pandas bar plots are categorical in nature. This means, bars are positionned at subsequent integer numbers, and each bar gets a tick with a label according to the index of the dataframe.
For example:
import matplotlib.pyplot as plt
import pandas as pd
s = pd.Series([30,20,10,40], index=[1,4,5,9])
s.plot.bar()
plt.show()

Here, there are four bars, the first is at positon 0, with the first label of the series' index, 1. The second is at positon 1, with the label 4 etc.
In contrast, a matplotlib bar plot is numeric in nature. Compare this to
import matplotlib.pyplot as plt
import pandas as pd
s = pd.Series([30,20,10,40], index=[1,4,5,9])
plt.bar(s.index, s.values)
plt.show()

Here the bars are at the numerical position of the index; the first bar at 1, the second at 4 etc. and the axis labelling is independent of where the bars are.
Note that you can achieve a categorical bar plot with matplotlib by casting your values to strings.
plt.bar(s.index.astype(str), s.values)
The result looks similar to the pandas plot, except for some minor tweaks like rotated labels and bar widths. In case you are interested in tweaking some sophisticated properties, it will be easier to do with a matplotlib bar plot, because that directly returns the bar container with all the bars.
bc = plt.bar()
for bar in bc:
bar.set_some_property(...)