[Regarding Python 2 - in Python 3 just use normal strings that are always Unicode.] // quick example of encoding and decoding a international domain name in Python (from Unicode to Punycode or IDNA codecs and back). Pay attention to the Unicode versus byte strings

quick_punycode_encode_decode_example.py

# INCORRECT! DON'T DO THIS!
>>> x = "www.alliancefrançaise.nu"  # This is the problematic line. Forgot to make this a Unicode string.
>>> print x
www.alliancefrançaise.nu
>>> x.encode('punycode')
'www.Alliancefranaise.nu-h1a31e'
>>> x.encode('punycode').decode('punycode')
u'www.Alliancefran\xc3\xa7aise.nu'
>>> print x.encode('punycode').decode('punycode')
www.alliancefrançaise.nu
>>> print x
www.alliancefrançaise.nu
>>> x == x.encode('punycode').decode('punycode')
/usr/bin/ipython:1: UnicodeWarning: Unicode equal comparison failed to convert both arguments to Unicode - interpreting them as being unequal
  #!/usr/bin/env python
False

# CORRECT FOR PUNYCODE (ALMOST THE BEST):
>>> x = u"www.Alliancefrançaise.nu"  # The difference! The Unicode string (decoded) string must be Unicode type
>>> print x
www.alliancefrançaise.nu
>>> x.encode('punycode')
'www.alliancefranaise.nu-dbc'
>>> x.encode('punycode').decode('punycode')
u'www.alliancefran\xe7aise.nu'
>>> print x.encode('punycode').decode('punycode')
www.alliancefrançaise.nu
>>> x == x.encode('punycode').decode('punycode')
True

# BEST ('idna' is preferable to 'punycode', see http://en.wikipedia.org/wiki/Punycode and https://docs.python.org/2/library/codecs.html#module-encodings.idna ) :
>>> x = u"www.alliancefrançaise.nu"
>>> print x
www.alliancefrançaise.nu
>>> x.encode('idna')
www.xn--alliancefranaise-npb.nu
>>> x.encode('idna').decode('idna')
u'www.alliancefran\xe7aise.nu'
>>> print x.encode('idna').decode('idna')
www.alliancefrançaise.nu
>>> x == x.encode('idna').decode('idna')
True

thx!

The third example, using idna encoding, shows the codec applying conversion to lower-case. I also get that with Python 3.6. However the final equality test then fails, for the obvious reason that www.Alliancefrançaise.nu does not equal www.alliancefrançaise.nu.
I'm curious why you got the result True for this.

x
Out[11]: 
'www.Alliancefrançaise.nu'
print(x.encode('idna').decode('idna'))
www.alliancefrançaise.nu
x == x.encode('idna').decode('idna')
Out[13]: 
False

Pay attention that in python3.7+ encode('idna') converts the string to lowercase, so:

>>> x = u"www.Alliancefrançaise.nu"
>>> x == x.encode('idna').decode('idna')
False

thanks @tuck1s and @icamys, I updated it per your comments

Section two not updated by comments @tuck1s and @icamys

# CORRECT FOR PUNYCODE (ALMOST THE BEST):
>>> x = u"www.Alliancefrançaise.nu"  # The difference! The Unicode string (decoded) string must be Unicode type