I think second one, assign
is used if want nice code witch chaining all functions - one line code:
df = pd.DataFrame({'A':np.random.rand(10000)})
default_value = 10
In [114]: %timeit df_new = df.assign(new_column=default_value)
228 µs ± 4.26 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
In [115]: %timeit df['new_column'] = default_value
86.1 µs ± 654 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
I use perfplot for ploting:

import perfplot
default_value = 10
def chained(df):
df = df.assign(new_column=default_value)
return df
def no_chained(df):
df['new_column'] = default_value
return df
def make_df(n):
df = pd.DataFrame({'A':np.random.rand(n)})
return df
perfplot.show(
setup=make_df,
kernels=[chained, no_chained],
n_range=[2**k for k in range(2, 25)],
logx=True,
logy=True,
equality_check=False,
xlabel='len(df)')