I've Googled this to high heck with absolutely no luck. Every Dream video I've noticed that outdoors always has daylight levels for him, yet in caves it's normal. This is always exactly what I want as 100% fullbright seems too cheat-y to me. Yet, no matter how I phrase this, Google simply refuses to give me the mod he uses. Did HE MAKE this mod? "Fullbright day", "Always day skylight", "Fully lit skylight", "Bright nights", no matter how I phrase it, Google refuses to give me anything but 100% fullbright mods which is NOT what I'm looking for!
They probably use Optifine, which lets you create a custom lightmap, in this case, with sky light always at the maximum (at least for higher levels, as making 0 fullbright will affect dark caves) while block light is unaffected (you can even set the lowest level to pitch black if you want so dark caves are truly dark). It could even be this texture pack (comments even mention Dream, though one says they don't need torches while mining so it seems to affect block light as well, in which case you could simply edit options.txt and set gamma to 100 or something).
I don't know how to use Optifine's custom lightmaps but it sounds like there are templates included and you just need to edit the topmost row (sky light) to remove the dusk/dawn/night colors (make it all one color except for the rightmost column if you still want to see lightning flashes), the one included doesn't look like vanilla though (it doesn't turn red at sunrise/sunset, just darker):
In the top half, the left-hand side represents night and the right-hand side represents day, with the dusk/dawn transitions in between. The very far right of the palette represents lightning flashes. Again, there is no specified width for the palette, but more width means more room for detail in the transitions.