I stumbled upon this link in Wikipedia and found that,
The term [Majesty] was first assumed by Charles V, who believed that—following his election as Holy Roman Emperor in 1519—he deserved a style greater than Highness, which preceding emperors and kings had used. Soon, Francis I of France and Henry VIII of England followed his example.
I never thought Highness and Majesty actually mean two different things. What are the differences? Is this a strictly an European concept? I imagine this doesn't translate really well to, say, Chinese Emperors or perhaps even late Roman Emperors (as with the title Emperor and King, a frequent question in this sub).
I can't answer for China but most of the time the term Majesty ( or Your Majesty ) is the form of address of the ruling monarch and his/her consort. For example in the UK the Queen is referred as Her Majesty. When France was a monarchy, the King and the Queen were referred as Their Majesties.
This is a form of address much more recent than some may think, unless I am mistaken it came at the time of Holy Roman Emperor Charles V ( as you said ), Francis I of France and Henry VIII.
Generally, Majesty is only used for the most powerful ( in terms of rank ) person of the state/country ( like Queen Elizabeth ) which is not the same as Highness.
Highness is a courtesy title used for members of the Royal family but not the sovereign monarch. Princes and Princesses are often Royal Highness. This is mostly about differences in rank.
TL,DR : Majesty is used for kings, queens, emperors and empresses which are the highest ranks in a country. Highness is used for princes and princesses.
Today, this is a common concept, even if there are different protocols. The Emperor of Japan is addressed as His Majesty and not His Imperial Majesty. On the other hand, I don't think a Roman Emperor was ever called Your Majesty as this seem a too recent form of address, but someone will know more about how a Roman Emperor was addressed ( which is a question I'd like to be answered by someone ;) ).
I can answer for Italy, though. Sua Altezza Reale (S.A.R., "Your Highness") and Sua Maestà (S.M., or "Your Majesty") cannot be used interchangeably as they were/are employed when addressing different people - the first title was/is used to refer to any member of the Royal Family, while the latter was reserved to the ruling monarch.