[AngelicaDeAlquezar]

Both mean that it's daytime and that the sun is shining. But often, "hace sol" implies that weather is rather hot, and "hay sol" just means that there is bright light of the sun.

Aún hace sol; vamos a caminar a la playa. -> It's still sunny and warm, let's go walking at the beach.
Aún hay sol; vamos a caminar a la playa. -> There is still sunlight, let's go walking at the beach.

No hubo sol en todo el día. ≈ No hizo sol en todo el día. -> It was cloudy and cool and sun didn't shine.

No hace mucho sol; no necesitas ponerte gorra. -> It's not very sunny, you don't need to wear a cap.
No hay mucho sol; más bien hace un poco de frío. -> It's not very sunny, it's rather a bit cold.

Estuvo haciendo mucho sol y se me quemó la espalda. -> Sun was shining very warm and I got my back sunburnt.
Había mucho sol allá afuera y siento que aquí adentro está muy oscuro. -> It was very sunny out there and I feel it's very dark in here.


@ookami: I agree... "hace luna" doesn't sound right.