Delicatessen Store (“Deli”)

The "appy" and the "deli" both are associated with New York.

"Appetizing stores" sell fish and dairy products.

"Delicatessen stores" sell meats.


(Oxford English Dictionary)
delicatessen
orig. U.S.
[G. delikatessen, Du. delicatessen (see DELICATESSE).]
a. Delicacies or relishes for the table; esp. attrib., in delicatessen shop, store.
b. ellipt. A delicatessen shop.
[1877 E. S. DALLAS Kettner's Bk. of Table 399 A house which abounds in foreign dainties of all sorts Lingner's Delicatessen Handlung, 46, Old Compton Street, Soho.]
1889 Kansas Times & Star 7 Nov., Burglars broke into Blake's delicatessen store..and..made an awful mess of the juicy stuff, canned and bottled.

(Oxford English Dictionary)
deli
colloq. (orig. U.S.)
Abbrev. of DELICATESSEN, usu. in sense b.  
1954 J. A. WEINGARTEN Amer. Dict. Slang 99/1 Deli... I have not yet seen this word in print, but it is used quite frequently. 'I'm having deli tonight.'; 'Mom, let's have deli.' Perhaps to be spelled dely or delly.

27 March 1875, New York (NY) Times, pg. 3:
A few years since, August, it appears, dyed for a living, in Jersey, while Caesar ministered to the living in the form of sausages, sauerkraut, and other delicatessen.

January 1879, Lippincott's Magazine of Popular Literature and Science, pg. 104:
Lager is ubiqitous; inscriptions of Deutche Delicatessen invite us to a light repart of sauerkraut and potato-salad;...

6 March 1884, Jackson Sentinel (Maquoketa, Iowa), pg. 6, col. 6:
In the window of a "delicatessen" store on Third avenue yesterday there appeared the head of a large porker ornamented with flowers and vines in an extraordinary manner. (...) -- N. Y. Sun.

16 November 1884, New York (NY) Times, pg. 3:
A fire broke out at 12:15 A. M. yesterday in the delicatessen store of August Kaaratedt, on the first floor of the five-story tenement hourse No. 205 Tenth-avenue.

5 February 1885, Brooklyn (NY) Daily Eagle, pg. 3:
ON SALE -- BUSINESS CHEAP, grocery and delicatessen store. 381 Atlantic av.

29 March 1885, Brooklyn (NY) Daily Eagle, pg. 12:
No attempt is made to sell foods made or grown in this country, there sold being what the Germans would call a "delicatessen."