Etymology of buy from Dictionary.com:
before 1000; Middle English byen, variant of byggen, buggen, Old
English bycgan; cognate with Old Saxon buggjan, Gothic bugjan to
buy, Old Norse byggja to lend, rent
Etymology of purchase from Dictionary.com:
before 1150; (v.) Middle English purchasen < Anglo-French
purchacer to seek to obtain, procure ( Old French pourchacier ),
equivalent to pur- (< Latin prō pro1 ) + chacer to chase1 ; (noun)
Middle English < Anglo-French purchas ( Old French porchas ),
derivative of the v.
Assuming this is true, we surmise that the word purchase is "newer" to English, or that it came into usage more recently than buy. What major event happened between 1000 A.D. and 1150 A.D.?
The Norman conquest of England. 1066 A.D. One of only a handful of things that modern-day U.S. students learn in high school, second only to that Eli Whitney patented the cotton gin.
After this, there was a major cultural shift in the area. From the same Wikipedia article:
One of the most obvious effects of the conquest was the introduction
of Anglo-Norman, a northern dialect of Old French, as the language of
the ruling classes in England, displacing Old English. French words
entered the English language, and a further sign of the shift was the
usage of names common in France instead of Anglo-Saxon names. Male
names such as William, Robert and Richard soon became common; female
names changed more slowly. The Norman invasion had little impact on
placenames, which had changed significantly after earlier Scandinavian
invasions. It is not known how much English the Norman invaders
learned, nor how much the knowledge of French spread among the lower
classes, but the demands of trade and basic communication probably
meant that at least some of the Normans and native English were
French and Latin became the only allowed languages in courts of law. French and Romance culture in general reigned as the more "civilized" and "advanced" society. Old English speakers and Old English words were shunned and ridiculed as the language of the peasants.
Cut to present day, we still see this in effect in today's English. To say that we have a gut-ache, would sound practically vulgar. Instead, we use the Romance word, stomache-ache.
Origin: 1300–50; Middle English stomak < Latin stomachus
gullet, stomach < Greek stómachos orig., opening; akin to stoma
You don't pay for things with a writ, you pay for them with a check/checque.
Origin: before 900; Middle English, Old English; cognate with Old
Norse rit writing, Gothic writs letter. See write
In England and America, we don't eat swine — although people who speak Germanic languages still eat Schweinfleisch. No no, we eat porque. Not swine. Swine sounds gross.
Oh yeah! Gross! Isn't that funny, considering what "gross" typically means in American and British English. It's used to describe something repulsive, whereas in German it just means... big. However, if we want to talk about something being large:
1125–75; Middle English < Old French < Latin larga, feminine of largus ample, generous
So in conclusion, I would say that yes, purchase is more "formal" than buy, especially in legal and intellectual matters, simply because of the thousand-year-old cultural cataclysm I described.