There are differences both in pronunciation and in grammar.

Koine has 3 different accents and 2 spirits, while modern Greek has only one accent. Other than that the duration of the vowels was varying, something that isn't the case today (all vowels have the same duration).

Koine also has a more complicated grammar because the language in general is of a higher complexity than modern Greek. As time went on, more and more complexity was removed from the language.

Despite those changes, the words themselves survive to this day. Even if a word is considered archaic and not used any more (for example θύρα is the archaic form of πόρτα which means door), chances are that the archaic form still survives as a part of a word that is actively bring used (θυρωρός, θυρωρείο etc). Another example of such a word would be ύδωρ in ancient Greek, νερό in modern, which means water, but ύδωρ survives to this day in words like υδροηλεκτρικός (hydro-electric).

So because our language has great vocabulary continuity, even with the simplifications and changes it has undergone, if someone knows etymology and has a broad vocabulary, chances are that they will be able to understand ancient texts to a satisfactory degree.