These do two completely different things, so you cannot compare them directly. This is easily verifiable:

In [2]: import numpy as np, pandas as pd

In [3]: df = pd.DataFrame(data=np.random.randint(0,2, size=(10,3)), columns=('a','b','c'))

In [4]: df
Out[4]:
   a  b  c
0  0  1  0
1  1  1  1
2  1  1  0
3  1  1  1
4  0  1  1
5  0  0  0
6  1  0  1
7  0  0  0
8  1  1  0
9  1  0  1

In [5]: df.any()
Out[5]:
a    True
b    True
c    True
dtype: bool

In [6]: any(df)
Out[6]: True

pandas.DataFrame.any is a method that does an "or" reduction operation across some dimension (by default, the 0th axis) which results in some pandas.Series object. In contrast, the built-in any takes an iterable, and does this reduction on an iterable. The result is always a bool object. When you iterate over a pandas dataframe, you iterate over the columns. So for the above df, the operation any(df) is equivalent to:

In [8]: list(df)
Out[8]: ['a', 'b', 'c']

In [9]: any(['a', 'b', 'c'])
Out[9]: True

Again, note, you can choose the axis for the .any method, like most methods in pandas:

In [10]: df.any(axis=1)
Out[10]:
0     True
1     True
2     True
3     True
4     True
5    False
6     True
7    False
8     True
9     True
dtype: bool

Note, if you worked with a pd.Series, which iterates over the values in the series, the operation would be (almost) the same:

In [12]: any(df['a'])
Out[12]: True

In [13]: all(df['a'])
Out[13]: False

In [14]: df['a'].any()
Out[14]: True

In [15]: df['a'].all()
Out[15]: False

Barring how in vanilla Python, float('nan') is treated as truthy, whereas in pandas by default, are skipped.

However, you should use the pandas methods for pandas data structures, because they are heavily optimized.

To compare two pd.DataFrame objects for both content and structure equality you can use:

import pandas as pd

def are_df_equal(df: pd.DataFrame, df2: pd.DataFrame) -> bool:
    return df.equals(df2) and (df.all() == df2.all()).all()