Read here, if you arrived here looking read about the difference between 'string' and object dtypes in pandas. As of pandas 1.5.3, there are two main differences between the two dtypes.
1. Null handling
object dtype can store not only strings but also mixed data types, so if you want to cast the values into strings, astype(str) is the prescribed method. This however casts all values into strings, even NaNs become literal 'nan' strings. string is a nullable dtype, so casting as 'string' preserves NaNs as null values.
x = pd.Series(['a', float('nan'), 1], dtype=object)
x.astype(str).tolist()          # ['a', 'nan', '1']
x.astype('string').tolist()     # ['a', <NA>, '1']
A consequence of this is that string operations (e.g. counting characters, comparison) that are performed on object dtype columns return numpy.int or numpy.bool etc. whereas the same operations performed on 'string' dtype return nullable pd.Int64 or pd.Boolean dtypes. In particular, NaN comparisons return False (because NaN is not equal to any value) for comparisons performed on object dtypes, while pd.NA remains pd.NA for comparisons performed on 'string' dtype.
x = pd.Series(['a', float('nan'), 'b'], dtype=object)
x == 'a'
0     True
1    False
2    False
dtype: bool
    
    
y = pd.Series(['a', float('nan'), 'b'], dtype='string')
y == 'a'
0     True
1     <NA>
2    False
dtype: boolean
So with 'string' dtype, null handling is more flexible because you can call fillna() etc. to handle null values however you want to.1
2. string dtype is clearer
If a pandas column is object dtype, values in it can be replaced with anything. For example, a string in it can be replaced by an integer and that's OK (e.g. x below). It might have unwanted consequences afterwards if you expect each value in it to be strings. string dtype does not have that problem because a string can only be replaced by another string (e.g. y below).
x = pd.Series(['a', 'b'], dtype=str)
y = pd.Series(['a', 'b'], dtype='string')
x[1] = 3                        # OK
y[1] = 3                        # ValueError
y[1] = '3'                      # OK
This has the advantage where you can use select_dtypes() to select only string columns. In other words, with object dtypes, there is no way to identify string columns, but with 'string' dtypes, there is.
df = pd.DataFrame({'A': ['a', 'b', 'c'], 'B': [[1], [2,3], [4,5]]}).astype({'A': 'string'})
df.select_dtypes('string')      # only selects the string column
    A
0   a
1   b
2   c
df = pd.DataFrame({'A': ['a', 'b', 'c'], 'B': [[1], [2,3], [4,5]]})
df.select_dtypes('object')      # selects the mixed dtype column as well
    A   B
0   a   [1]
1   b   [2, 3]
2   c   [4, 5]
3. Memory efficiency
String Dtype 'string' has storage options (python and pyarrow) and if the strings are short, pyarrow is very efficient. Look at the following example:
lst = np.random.default_rng().integers(1000000, size=1000).astype(str).tolist()
x = pd.Series(lst, dtype=object)
y = pd.Series(lst, dtype='string[pyarrow]')
x.memory_usage(deep=True)       # 63041
y.memory_usage(deep=True)       # 10041
As you can see, if the strings are short (at most 6 characters in the example above), pyarrow is consumes over 6 times less memory. However, as the following example shows, if the strings are long, there's barely any difference.
z = x * 1000
w = (y.astype(str) * 1000).astype('string[pyarrow]')
z.memory_usage(deep=True)       # 5970128
w.memory_usage(deep=True)       # 5917128
1 Similar intuition already exists for str.contains, str.match for example.
x = pd.Series(['a', float('nan'), 'b'], dtype=object)
x.str.match('a', na=np.nan)
0     True
1      NaN
2    False
dtype: object