As someone who grew up a hardcore spinning rod, minnow guy, and still tosses a spinning rod around now and then.
The short answer is, it depends on the situation.
For me, a fly rod gets me a better drift and is more accurate. For spooky or parsnickety fish. It also enables dry flies, which, in certain situations will outfish underwater options (yes, a trout may generally feed subsurface for 90% of its food intake, but, its about when. 10% of the time they feed almost exclusively topwater, we can target those times, and when it happens, its easier, because they tell you exactly where they are and what they're eating!) A fly rod is also generally superior on small streams, because you can more effectively fish shallow riffs and tiny pockets, rather than being relegated to pools only.
A spinning rod is superior in bigger water where longer casts are necessary, getting to depths over 6 ft or so, as well as for aggressive fish where working something fast (like spinners) is effective and covering huge swaths of water quickly is advantageous.
Just my 2 cents.