The Emperor's hoodie was, at least according to the Star Wars: Visual DictionaryStar Wars: Visual Dictionary, intended to hide his face from his followers.
And in the AotC novelisationAttack of the Clones novelisation, it's apparent that Obi-Wan uses his hood to cover his face from the elements:
At last Obi-Wan put his fighter down on the designated landing pad. He pulled pulled his hood up tight over him, then slid back his canopy and scrambled scrambled out against the wind and the rain, rushing across the permacrete permacrete to a tower across the way. A door slid open before him, spilling spilling out brilliant light, and he went through, crossing into a brightly brightly lit white room.
“Master Jedi, so good to see you,” came a melodic voice.
Obi-Wan pushed back his hood, which had offered little protection from the the driving rain, and brushed the water from his hair. Wiping his face face, he turned to face the speaker, and then he paused, caught by the image image of the Kaminoan.