It is rather subjective to say whether there are “advantages” to each method.

However, a good understanding of what goes under the hood would make it natural
for one to pick the best choice for each occasion.

A decorator (talking about function decorators), is simply a callable object that takes a function as its input parameter. Python has its rather interesting design that allows
one to create other kinds of callable objects, besides functions – and one can put that to use
to create more maintainable or shorter code on occasion.

Decorators were added back in Python 2.3 as a “syntactic shortcut” for

def a(x):
   ...

a = my_decorator(a)

Besides that, we usually call decorators some “callables” that would rather be “decorator factories” – when we use this kind:

@my_decorator(param1, param2)
def my_func(...):
   ...

the call is made to “my_decorator” with param1 and param2 – it then returns an object that will be called again, this time having “my_func” as a parameter. So, in this case, technically the “decorator” is whatever is returned by the “my_decorator”, making it a
“decorator factory”.

Now, either decorators or “decorator factories” as described usually have to keep some internal state. In the first case, the only thing it does keep is a reference to the original function (the variable called f in your examples). A “decorator factory” may want to register extra state variables (“param1” and “param2” in the example above).

This extra state, in the case of decorators written as functions is kept in variables within the enclosing functions, and accessed as “nonlocal” variables by the actual wrapper function. If one writes a proper class, they can be kept as instance variables in the decorator function (which will be seen as a “callable object”, not a “function”) – and access to them is more explicit and more readable.

So, for most cases it is a matter of readability whether you will prefer one approach or the other:
for short, simple decorators, the functional approach is often more readable than one written as a class – while sometimes a more elaborate one – especially one “decorator factory” will take full advantage of the “flat is better than nested” advice fore Python coding.

Consider:

def my_dec_factory(param1, param2):
   ...
   ...
   def real_decorator(func):
       ...
       def wraper_func(*args, **kwargs):
           ...
           #use param1
           result = func(*args, **kwargs)
           #use param2
           return result
       return wraper_func
   return real_decorator

against this “hybrid” solution:

class MyDecorator(object):
    """Decorator example mixing class and function definitions."""
    def __init__(self, func, param1, param2):
        self.func = func
        self.param1, self.param2 = param1, param2

    def __call__(self, *args, **kwargs):
        ...
        #use self.param1
        result = self.func(*args, **kwargs)
        #use self.param2
        return result

def my_dec_factory(param1, param2):
    def decorator(func):
         return MyDecorator(func, param1, param2)
    return decorator

update: Missing “pure class” forms of decorators

Now, note the “hybrid” method takes the “best of both Worlds” trying to keep the shortest and more readable code. A full “decorator factory” defined exclusively with classes would either need two classes, or a “mode” attribute to know if it was called to register the decorated function or to actually call the final function:

class MyDecorator(object):
   """Decorator example defined entirely as class."""
   def __init__(self, p1, p2):
        self.p1 = p1
        ...
        self.mode = "decorating"

   def __call__(self, *args, **kw):
        if self.mode == "decorating":
             self.func = args[0]
             self.mode = "calling"
             return self
        # code to run prior to function call
        result = self.func(*args, **kw)
        # code to run after function call
        return result

@MyDecorator(p1, ...)
def myfunc():
    ...

And finally a pure, “white colar” decorator defined with two classes – maybe keeping things more separated, but increasing the redundancy to a point one can’t say it is more maintainable:

class Stage2Decorator(object):
    def __init__(self, func, p1, p2, ...):
         self.func = func
         self.p1 = p1
         ...
    def __call__(self, *args, **kw):
         # code to run prior to function call
         ...
         result = self.func(*args, **kw)
         # code to run after function call
         ...
         return result

class Stage1Decorator(object):
   """Decorator example defined as two classes.
   
   No "hacks" on the object model, most bureacratic.
   """
   def __init__(self, p1, p2):
        self.p1 = p1
        ...
        self.mode = "decorating"

   def __call__(self, func):
       return Stage2Decorator(func, self.p1, self.p2, ...)


@Stage1Decorator(p1, p2, ...)
def myfunc():
    ...

2018 update

I wrote the text above a couple years ago. I came up recently with a pattern I prefer due to creating code that is “flatter”.

The basic idea is to use a function, but return a partial object of itself if it is called with parameters before being used as a decorator:

from functools import wraps, partial

def decorator(func=None, parameter1=None, parameter2=None, ...):

   if not func:
        # The only drawback is that for functions there is no thing
        # like "self" - we have to rely on the decorator 
        # function name on the module namespace
        return partial(decorator, parameter1=parameter1, parameter2=parameter2)
   @wraps(func)
   def wrapper(*args, **kwargs):
        # Decorator code-  parameter1, etc... can be used 
        # freely here
        return func(*args, **kwargs)
   return wrapper

And that is it – decorators written using this pattern can decorate
a function right away without being “called” first:

@decorator
def my_func():
    pass

Or customized with parameters:

@decorator(parameter1="example.com", ...):
def my_func():
    pass
        
        

2019 – With Python 3.8 and positional only parameters this last pattern will become even better, as the func argument can be declared as positional only, and require the parameters to be named;

def decorator(func=None, /, *, parameter1=None, parameter2=None, ...):