I don't think I've ever heard anyone say "boiling hot" to refer to the weather. If you're just talking about everyday conversation I wouldn't read too much into it--I would just assume that both meant "really hot."
"It's boiling hot" is common in BE (when it gets hot enough.) It is tempting to think that there is a connection between "boiling" and steam and humidity... but there isn't. To me "boiling hot" is primarily a weather (and hot drink) description, and "burning hot" is primarily the description of an object.
Approximately: boiling hot = as hot as boiling water; burning hot = hot enough to burn you.