Maybe the hero's power is generic. It's the same old 'holy sword' and 'light magic' every Chosen One gets. Your power is unique, tailored to your specific backstory and choices. A unique, bespoke skillset will always feel more potent and personal than an off-the-rack destiny package. The sense of ownership over your abilities makes them feel more powerful, even if the damage numbers are similar.
The assassin's guild, if one exists, would have a total meltdown. You're either their greatest marketing tool or their worst nightmare. Do they celebrate you and risk the wrath of the state? Or disavow you to maintain their low profile? Your treatment from your own profession would be deeply conflicted—pride mixed with profound anxiety about the heat you're bringing down on everyone.
Depth over breadth. The hero is widely known, but shallowly understood—a symbol. Your reputation is known to far fewer, but they understand its implications with devastating depth. A king who knows your name understands exactly what it means, down to his own mortality. That deep, personalized understanding of your capability is a more intense form of 'fame' than the cheers of a thousand strangers.
That time you saved the kingdom by quietly removing the corrupt chancellor weeks before the "hero" even arrived at the capital. The official chronicles credit a sudden illness, but we know the truth.
Okay, but has anyone actually counted the kill counts? I'm just saying, in a purely utilitarian sense, the assassin probably has a higher body count of meaningful targets. The hero clears mobs; the assassin removes key figures. Removing one kingpin does more to dismantle a criminal empire than beating up a hundred foot soldiers. By that metric of efficiency and impact, his 'power' output is higher.