I think there are several interpretations, with no definitive rules support one way or another, so it's dm judgement call.
No, flaming sphere stays lit, and pyrotechnics causes the fireworks or smoke
The flaming sphere stays lit and you get the fireworks/smoke.
The reason the sphere stays lit is that it is fueled by magic, not mundane fuel.
While there isn't explicit rules support for this interpretation, it isn't unbalanced because it isn't more powerful than the spells separately, and deciding in favor of the PCs and the rule of cool argue in favor of it.
Perhaps for fluff it goes out during the fireworks/smoke, then relights.
A Contrary Argument
A contrary argument would be that you get either spell, but not both. Basically either one spell or the other wins.
If pyrotechnics wins, then the flaming sphere is extinguished and you get your choice of the fireworks or smoke.
If flaming sphere wins, then pyrotechnics fails to extinguish the flame.
This is making the assumption that the fire in the flaming sphere is magically fueled for the duration, and it seems reasonable to assume that it can't be snuffed the same way as a regular fire, because magic.
So how to decide which spell wins? Assuming they're both cast by the same caster, it seems reasonable to let the caster choose. If cast by different casters in an antagonistic situation, if the spells have been cast at different levels, higher level wins; otherwise, its a contest between spellcasting abilities.
No matter what, the fires created by the flaming sphere are fair game
You can use the flaming sphere to ignite something, and then use pyrotechnics to blow that stuff up like a boss, or, create smoke, of course.
Again, there's not much in the way of rules support, but if you think allowing the sphere to stay lit is overpowered, then perhaps this is the way to go.
Note:
My original answer was significantly different from this but editing it made me re-think it.
Yes, you can move the image out of the original range
"Within range of the illusion" refers to the 20 ft cube illusionary object or creature, NOT the 120 ft casting radius.
As long as you are within 120 ft radius of the illusionary object, you may move it to any other point within range. This lets illusionists keep one persistent Major Image for any amount of time if they can keep using actions to keep it near them.
What defines "any other point within range" however, is up to discussion. It could mean you may move the object either 120 ft within the caster, or 120 ft within the illusionary object.
Best Answer
The target is fixed
Any area spell in which the target area moves is very explicit about that. For example, Antimagic Field is specified to move as the caster does.
In this case, no mention is made of the area of effect moving. Separately, it mentions different possible effects for this area. None of those mention changing the location, either.
On the plus side, this cube is 100 feet on a side. If you use the same battle mat I have and the usual 5' per square, that 20x20 squares will cover most of the mat. Since it's 100' tall as well, that's 1 000 000 cubic feet of air you're controlling. Moving the area around shouldn't be that much of an issue.