Toffy is an alternative spelling of toffee. By itself, toffee refers to “a type of confectionery made by boiling sugar (or treacle, etc) with butter or milk”. However, the commonly-used combination toffee-nosed is an adjective meaning “snobbish, condescending or aloof”. Wiktionary shows the combination's etymology as “Probably from toff”, a toff being “A person of the upper class- or high-class-pretence who usually communicates an air of superiority”.
Wiktionary shows no etymology for toff, but OED1 (1926) shows:
Toff vulgar. Also rarely toft. [Perh. a
vulgar perversion of TUFT, as formerly applied to
a nobleman or gentleman-commoner at Oxford.]
An appellation given by the lower classes to a
person who is stylishly dressed or who has a smart
appearance; a swell; hence, one of the well-to-do,
a 'nob' ...
Hence Toffish, Toffy adjs. like or characteristic
of a 'toff', stylish.
OED1 entries under the toffee, toffy heading refer only to candy. The etymology is obscure:
Toffee, toffy [Of uncertain origin:
app. orig. dialectal, and sometimes spelt tuffy,
toughy, as if named from its toughness; but the
earlier form is the northern TAFFY, q. v.]
The OED1 (1919) entry for taffy shows no etymology for its “candy” and “soft soap” senses:
Taffy¹ The earlier form of TOFFEE, now Scotch, North Eng., and American.
1. A sweetmeat made from sugar or treacle, with butter, etc. : see TOFFEE. ...
2. U. S. slang. Crude or vulgar compliment or flattery; 'soft soap'; blarney. ...
3. attrib. and Comb., as taffy stand, stick; taffy-join, a reunion of young people for the making of taffy to which each contributes. ...
Taffy² [An ascribed Welsh pronunciation of Davy or David, in Welsh
Dafydd.] A familiar nickname for a Welshman: cf. Paddy, Sawney, etc.
The half-dozen OED1 cites for Taffy¹ sense 1 date from 1817 to 1890, and the cites for Toffee, toffy from 1825 to 1896. The Great Vowel Shift occurred 1350–1700.