So basically reputation matters and so does deeds because they're a cornerstone of a Demon's powers.
As an example:
You are Lordly McEvil, a Maou with a heavy focus on shadow magic tricks, playing to your forte. You've snuck into the church of the Verdant Goddess in the middle of the night, and slowly begin to slay her faithful as they sleep, utilizing the shadows of their rooms as blades to end them.
No one who witnesses you survives the night. You are filled with Demonic Power, and can use it to become better at your shadow magic, to enhance your magic core further, or etc Maou powers. But you do not gain any Deeds for it, even as the bodies are discovered within hours when non-clergy arrive for their duties. No one knows it was you, so no Deed can be attributed.
If you let the lowly page-boy flee, he may spread tales of your all consuming shadow, that ripped the heads from his teachers and gave them no time to react. This means not only do you gather the Demonic Power, but an Infamous Deed relating to your shadow magic may be made, making your shadow blades sharper in the future or such.
But imagine you killed everyone but wanted to create a Deed as well; If you leave a calling card that names you as the one who has done it, the deed may also spread; but you might be praised as a terrifying Assassin instead, and find the arts of daggers and stealth are now a part of your Myth, because no one knows the truth of what cut through these clergymen, giving you new capabilities you did not have rather than enhancing what you do.
