Because FYROM wants to change its name and Greece doesn't.
Macedonians of Greece object to being called South Macedonians all of a sudden because that's not their identity - as far as they're concerned they live in a region called Makedonia and consider themselves Makedones, and have been doing so for as long as anyone cares to remember.
Not to mention that most other Greeks perceive this as an attack on and usurpation of their national identity. Both countries fly the Vergina Star and claim Alexander the Great as their hero and ancestor. What is at stake here is the claim to the ancient legacy, not some Byzantine or Roman region.
Finally, South Macedonia happens to be most of historical Macedonia by today's borders, and most of it is located in Greece. Greeks find it extremely difficult to accept a naming change either way when they hold most of the territory with the contested legacy, and when their neighbours to the north have none of the historically important sites.
Any notion of "backing down" will not be received well at home, where sentiments are running high. The current Prime Minister, Alexis Tsipras, knows that the country is in a precarious position where nobody likes them at the moment, and being inflexible on the naming issue won't win him any favours abroad, so he has a very delicate balancing act to perform: to play with a weak hand internationally, while having the minimum amount of people calling him a traitor at home.
Additionally Tsipras' party is left-wing pro-European and it has to maintain its pro-EU stance even at the face of a negative outcome, in a country where Europe is not thought of in kind terms at the moment.
Finally, Greeks are aware that the outcome will set a precedent for other neighbouring states to claim their piece of their homeland. The most likely candidates are Albania and Turkey. Albania will want to claim parts of Epirus, and Turkey will of course go for Thrace. This is not at all beyond the pale as Johannes Hahn, a member of some European committee, is already talking about restructuring the Greco-Albanian border and Turkey seems more agitated than usual as of late.
Bulgaria, historically an antagonist, is quite friendly these days and is supporting Greece's stance on the naming issue, as they too find themselves facing territorial claims from their common neighbour that stem from the same nationalist idea which sparked the naming debate in the first place. Serbia is also considered an ally, largely due to Orthodox Christianity, and Greek support during the Kossovo war.
Apart from those two nations, Greece doesn't have very many friends at the moment, and sign of weakness will be paid for by future generations.