The phrase for show and not for blow and its variants refer to a handkerchief used as a clothing accessory, not for blowing the nose. The earliest occurrence that I have found is from an article titled On the Handkerchief Scene in Othello, published in The Morning Chronicle (Sydney, New South Wales, Australia) of Saturday 10^th February 1844:

It would seem rather strage [sic], that Othello should have had a pocket-handkerchief given to him at all; but it is probable that such an article was kept by him (as the boys say) “for show and not for blow;” so that the curiously-worked pattern, consisting, as it did, of strawberries worked on a white ground, made the celebrated handkerchief in question, just the sort of article that the Moor would have had given him.

Likewise, the phrase one for show and one for blow and its variants refer to two handkerchiefs, one used as a clothing accessory, the other for blowing the nose. These are the earliest occurrences of the phrase that I have found:

1-: From the Green Bay Advocate (Green Bay, Wisconsin, USA) of Thursday 19^th April 1877:

“One for show and one for blow,” said an Oshkosh young lady the other evening as she pulled two handkerchiefs out of her pocket for use about her nose.

2-: From The Hiawatha World (Hiawatha, Kansas, USA) of Thursday 21^st December 1882:

Every one carries two handkerchiefs, one for show and one for blow, these days.

The phrase for show and not for blow has come to be used figuratively to mean for display rather than for use. The earliest occurrence that I have found of this figurative use is from “Mid Pleasures and Palacs [sic]”, by ‘W. A. W.’, published in The Emporia Gazette (Emporia, Kansas, USA) of Saturday 29^th May 1909:

The only thing that costs more in Italy than in America is the room with a bath. Except in the more gorgeous hotels in the large cities, where the rates run from $5 a day up, there are no rooms with baths in Italian hotels. […] In every hotel on each floor is a big porcelean bathtub. It can be rented for 60 cents. What they call sitz bathtubs that look like a compromise between a sedan chair and a horse-trough and work like the washtub in the kitchen Saturday with the papers pinned to the windows against the neighbors—a sitz bath may be had in your room, cold for 2 cents and hot for 5 cents  So, if a man really desires to keep clean in Italy, he can do so at a reasonable figure. But if he is like most Americans who have their porcelean tubs merely for show and not for blow, to appear clean whether they are clean or not, it will cost about $6 a day to keep up the hollow pretense!

And the phrase one for show and one for blow has come to be used figuratively to mean one for display and one for use. The earliest occurrence that I have found of this figurative use is from an article about the trial in Indianapolis of members of the Iron Workers Union for involvement in the “dynamite conspiracy” case, published in the Meriden Morning Record (Meriden, Connecticut, USA) of Friday 25^th October 1912—here, the phrase is applied figuratively to something used as an appearance to hide the real character of an illegal activity:

According to one witness whose reliability seemed unquestioned, the iron workers had two sets of books “one for show and one for blow.”

An interesting early variant of the phrase for show and not for blow occurs in the following paragraph from the Democrat and Chronicle (Rochester, New York) of Wednesday 4^th July 1888:

Some of our Democratic friends affect to be horrified by the thought that the Republicans will put their handkerchiefs of stars and stripes to an ignoble use. They need not be alarmed. “Old Glory” is for waving, and not for “blowing.” And that is where it differs from the red bandanna.

What the above-quoted paragraph referred to had been explained in The Stars and Stripes Versus the Red Bandanna, published in the Democrat and Chronicle (Rochester, New York) of Thursday 14^th June 1888:

When Thurman ¹ was nominated, General Spinola ² tied a bandanna over the back of his desk, which remains there. To-day Henry Cabot Lodge ³ countered on Spinola by placing on his desk a brilliant United States flag about the size of a bandanna, which action was seconded and indorsed by Messrs. Reed, Burrows, Cannon and McKinley ⁴, all wearing silk flag handkerchiefs. It is a singular fact that the Democrats look upon the badge of the Republicans—the United States flag—as a hostile gonfalon. It remains to be seen whether the country will indorse the party of the bandanna or the party of our country’s flag.

¹ Allen G. Thurman (1813-1895) was the Democratic Party’s nominee for Vice President of the United States in 1888.
² The Democratic politician Francis B. Spinola (1821-1891) was a member of the U.S. House of Representatives from New York from 1887 to 1891.
³ Henry Cabot Lodge (1850-1924) was a Republican politician.
⁴ The Republican politicians Thomas B. Reed (1839-1902), Julius C. Burrows (1837-1915), Joseph G. Cannon (1836-1926) and William McKinley (1843-1901) were at that time members of the U.S. House of Representatives. (William McKinley was the 25^th President of the United States from 1897 until his assassination in 1901.)