Why do ppl say do it quickly but do it fast?
Native speaker mistake, or am I the one mistaken?

I haven't heard that expression before, and I can't find any reference to it on Google or Yahoo.
In what country are you hearing this?"

I've heard: "More speed less haste"

I suspect it's humorous, like saying: Do it quickly but do it QUICKLY!

I meant as two separate phrases, not as a saying.
Do you know the answer?

Why do people say: Do it fast (and not fastly)
Why do people say: Do it quickly? (and not quick)?

Why an adverb in one, and an adjective in the other (or vice versa)?

Ahhhh. Now I understand.

As you note, 'quickly' is an adverb and used correctly. We can usually tell that a word is an adverb because of the –ly ending.

HOWEVER, not all adverbs do, and ‘fast’ is one of them. Here are some more:
always, often, sometimes, seldom, and never, here, very, now, soon.

Some words can be both adverbs and adjectives - and 'fast' is one of them:
a fast car (adjective)
"He was driving too fast." (adverb)

Others include:

early, daily, weekly, and hourly, far
and
right :
“the right answer” (adjective )
“He was standing right behind her (adverb)
“I’ll be right back.” (adverb)

near:
“His retirement was drawing near.” (adverb)
“A a near perfect fit.” (adverb)
“in the near future” (adjective)

Thanks! That really helps