Sam, as far as weather data, can't help you there, but having been to both mountains many, many times, I can only say that while it can be clear on Whitney, it's storming badly on White Mtn and vice versa. they are approximately 60-70 miles apart on opposite sides of the Hwy 395 corridor in two different mountain ranges. Since the Whites are brown, black, red and green, I'm thinking they call them the Whites because they seem to get more snow than the High Sierra.
Whitney seems to warn you for a longer time it will storm, but White Mountain's storms seem to come in suddenly without warning. There's almost always high winds especially the closer to the summit you get. I think they draw out of the hot Crowley Lake Valley and straight up White's west side. A couple times I had to literally crawlon hands and feet on the narrow trail area down from the summit to keep from blowing off.
We got caught in monster hail/thunder storm at Barcroft Lab Open House of 2005. Since you're from down south and not familiar, Open House is first weekend of August (height of monsoon season) and will cut about 2.5 miles off the trail mileage. If you're not familiar with this event, go to
www.wmrs.edu I believe or search it. We got to the 14,000' level and it was clear until about 10:30 AM and the clouds just suddenly came out of nowhere. Got extremely cold and wet and lightning strikes were hitting way too close.
If you haven't been there, very worth the trip.