Would this be useful for war?
Probably.....who am I kidding, HELL YES!
Don't forget that this was a time when people thought that only the gods could fly in the heavens. Of course, the more educated would see right through this but it is not the more educated fighting the battles, it is the superstitious common person. If Rome could build hundreds or thousands of these she could simply scare most of her enemies into submission. It would also appear to her enemies that she can read the minds of their generals because of....
Roman generals would utilize this to fly over enemy encampments to get a better idea of what they were up against. They would simply hop into one and fly so far up that the enemies arrows couldn't reach them. They would get a better idea of how many enemies they were facing as well as the troop distribution of the enemy soldiers. This would make the following battle....
Imagine you are an ancient Carthaginian soldier fighting a battle against Rome. You are just waking up to the screams of your comrades. You exit your tent and see hundreds of your fellow Carthaginian soldiers running around with burning black goo on their faces and dozens more laying on the ground, passed out from misery with giant blisters on their faces.
You look up and see hundreds of balloons over the camp, dropping vats of boiling oil onto people as well as firing arrows and throwing rocks. The heavens have turned against you! Then, Roman Auxiliary cavalry and thousands of Legionnaires pour into the camp and begin to butcher your surprised, suffering and panicked brethren. Most of the guys in your squad are too hurt to fight back. This drives you and the remainder of your forces to the cities for a drawn out....
The Romans surround your city and you think they are preparing for a drawn out siege. What you don't know is that during the night the Romans have a surprise for the sleeping soldiers.
After you and your guys go to sleep, the Romans bring about thirty balloons over a deserted part of the city. They drop about a hundred special ops guys into the city who proceed to kill the guards open the gates silently. The entire Roman army pours in. You are beheaded in your sleep.
Would this be useful for war? Would this influence Rome's cartography enough to cause a breakthrough?
The Romans warred with the Persians for a long time because the Persians were wealthy. Now, throw these balloons into the equation and assume that the Persians haven't managed to figure out how to build then yet and Rome could conceivably conquer all of Persia and even parts of India and a few outposts in China.
They would likely end up having frontiers described in the next map.
They conquered Scotland and retook Mesopotamia from the Parthians.
What would be the consequences for Rome's navy?
Well, taking the previous into account Rome would likely build a type of aircraft carrier.