I'd go for Drought because it's way more useful. But for the heck of it, I picked both.
Drought = No charge for SolarBeam and it weakens Water attacks while strengthening Fire, which is really useful for Groudon against, say, Kyogre.
Heavy Rain = 100% accuracy for Thunder and strengthens Water attacks while weakening Fire; not really useful, but it still puts up some sort of challenge against the same type, especially if Kyogre knows Thunder.
How is Drought useful against Kyogre, but
Drizzle isn't useful against Groudon?
They're both equally useful. Drought allows for Solarbeam to be used without charging, Drizzle allows for Thunder to have 100% accuracy, and Hail allows for Blizzard to have 100% accuracy (just thought I'd add that.)
I pick Drizzle. The reason being is because it causes rain which strengthens water types. Since Kyogre is Water already, its Water attacks get stab. So basically, Kyogre's Water attacks are crazy strong (more than Groudon's Fire attacks.)
But, since Groudon is Ground type and not Fire type, its Fire attacks do NOT get stab, they only get one power up from Drought.
Also, Groudon has a high Attack stat. Kyogre has a high special defense stat. That means, when Groudon is using Solarbeam, it's weaker due to his weaker Sp.Attack stat and it's even weaker due to Kyogre's strong Sp.Defense stat.
And don't forget about the fact that Water is super effective against Ground.
In conclusion - Kyogre owns Groudon in every way.