There seems to be no in game answer to this question, so one may only guess: maybe there are some hidden requirements for using the Phoenix Down item (or any other Materia that could have the same effect. I wonder what would have happened if Aerith had Final Attack + Phoenix slotted in on her armor while she was praying...)? Some argue that during battles characters aren't killed but just "knocked out" (FF7 itself marks in-battle dead characters as KO'ed, but one should check the original Japanese wording to be sure), some other prefer to think that Phoenix down is useless when the wounds are too serious OR too much time has passed (which still isn't exactly coherent with the in game mechanics) but no official statement seems to have been made on the matter.
Each way, the whole "Aerith's death" incident is crucial to the plot (while maybe not as crucial as Squaresoft like to portrait it - I still believe that if it wasn't for added drama the plot could have worked out something different) so the whole scene is clearly to be taken as a classical example of story / game mechanics segregation: you should be able to revive a character using a resurrection item, but the plot need you not to be able too. Sadly, in this specific case it also fail at explain why.
That said, I wanted to point out that this is not the first case of such segregation in the Final Fantasy series. Following are just some examples that come to mind:
Back in Final Fantasy 4, some characters I am not going to name will get turned into stone at some point of the game. This is an important part of the plot, and for story purpose it is pretty important that they stay statues. After their death, another character tries to break the spell, only to fail "because they wily turned themselves to stone". This is reinforced by the game offering the player the ability to try to use an item on the stoned characters - any item will produce a generic "No effect" message but Golden Needle (an item only present in the Japanese release of the game and allegedly removed to avoid to confuse the players just because of this event) which is used to remove the Petrified status in battle produces a different "This can't work because the characters sacrificed themselves and turned into stone by their will" message. All fits so far... Until the characters are restored near the end of the game by another mage who breaks the curse after the game stated that it couldn't be broken when the first mage (who should be more powerful than the latter one, at least plot wise) tried the same thing... Such event can only really be explained with "plot-needs" and "characters-too-good-to-be-used-all-the-game".
Another character in Final Fantasy 5 follows the same destiny and dies after single handed facing a powerful enemy. The rest of the party tries to use magic to heal him (and I seem to remember that they try a curative item - maybe even Phoenix Tail / Down too) only to fail because "he is too dead for magic/items to work".
Following all above examples and for sake of an in-game explanation, I am going to suggest that Phoenix Down doesn't have the power to revive a killed character in some extreme situations (to much time passed, the body is too damaged, external factors keeps the soul from returning etc) and Aerith death should be considered one of such cases.
Maybe the planned PS4 remake will fix this (small) plot segregation. I personally would love to see the other characters trying to revive her only to fail for some reason they cannot understand, until the end of the game when they discover that she can't come back because she casted Holy to save the planet and her souls was busy in the Lifestream all the time...
Let's just hope that they won't go down the FF4 rout instead and just remove Phoenix Down from the game :P