In many varieties of English, a W is a W. In these varieties, W sounds like [w] like in ‘wise’ and ‘wonderful’ and ‘wowza’ (unless it’s at the end like in ‘draw’ or ‘stow’, in which case it’s quiet as a mouse.)
However, in Scottish, Irish, New Zealand, and certain American dialects, wh-words are pronounced a little different. In words like ‘which’ and ‘whale’, the H makes the W kind of…H-y.
Why is that? Why are those words even spelled with an H to begin with? As with many questions about the bitter rivalry between English pronunciation and English spelling, we have to look to the distant past…
Or, you know, the fairly old past—the Old English-y one. Old English inherited from Indo-European (with a few twists and turns through Grimm’s Law) a sound we linguists like to call a voiceless labiovelar approximant1. In IPA, [ʍ]. That’s fancy language-people talk for a kind of voiceless W. In OE, this sound was spelled ‘hw’. ‘Which’ was ‘hwilc’ and ‘whale’ was ‘hwæl’. Perhaps the real poster child for this phenomenon is the first word of Beowulf: Hwæt! (ModE ‘what’)
During the Middle English period, the spelling of this sound was flipped to our modern ‘wh’, most likely due to the influence of French scribes who came to England with the Normans. It was also sometime during this period that some dialects began to see a merger between the pronunciation of ‘wh’ and plain old ‘w’. For a while, the merger was seen as uncouth, and educated speakers deliberately maintained the [ʍ] pronunciation of ‘wh’. Now, we find more dialects than not where the merger is complete and both spellings are pronounced [w]. But as mentioned before, there are several varieties of English where the original [ʍ] is hanging in there.
English has its share of strange, purely historic spellings, but this isn’t one of them. Your [ʍ] dropping friend isn’t mispronouncing ‘white’ or being pedantic; they’re just kicking it old school.
1This sound is sometimes traditionally/erroneously called a labiovelar fricative.