Sometimes people say
"Er hat Grippe."
and sometimes they say
"Er hat Männergrippe."
When to use which? Where is the difference?
Like πάντα ῥεῖ wrote, "Männergrippe" is not a real illness but a joke.
This joke bases on the assumption, men are more histrionics if they get ill than women.
In some comedy the man is shown helpless and whining because he has a cold. The woman instead organizes whole household, children and her job with a broken leg.
So the use of "Männergrippe" let you know, your conversational partner has doubts about the severity of the illness.
Then basing on the mimic you can see, if this person is joking, or really upset about this overstatement of the ill/not so ill person.
In difference "Grippe" is a real disease. It spreads fast. Sometimes, if it is wide spread one talks about "Grippewelle", and there exists vaccinations against it.
The word "Männergrippe" is used mockingly by women who perceive that men complain and whinge too much when they have a flu or simply a cold. The expression is not exclusively German, you can find "man flu" in English language dictionaries:
[..] (informal, humorous)
a cold or similar minor illness that a man catches and treats as if it were flu or something more serious
- Greg was off sick with man flu, according to his wife.
(Source: Oxford Learners' Dictionaries)
Actually, it seems that men suffer indeed more from a flu than women. This may be due to the way in which the sexual hormones testosterone and oestrogen affect the immune response (see Infektionen: "Männergrippe" ist kein Mythos or Man flu is real because oestrogen protects women from the influenza virus).
In a more scientific article: The science behind “man flu” by author Kyle Sue you can read:
The concept of man flu, as commonly defined, is potentially unjust. Men may not be exaggerating symptoms but have weaker immune responses to viral respiratory viruses, leading to greater morbidity and mortality than seen in women.
Overall, evidence seems to be limited.
In addition to all the other entries here. I can say that when I have a "Grippe" I often refer to it as a "Männergrippe" to tell others that I am very very sick. In a joking way and with additions in "please call an ambulance for me" and so on. Very much jokink and "mimimi". If a guy tells another guy he has a "Männergrippe" we often ask If we should call an ambulance too (jokingly).