Pro: Unifies string and symbol syntax, which are almost the same anyway. Frees up | and \ for other uses. Easy to implement. See also #360.
Con: Nobody implements or uses this, whereas |...| is widespread in R5RS systems and \ is required in R6RS systems.
Racket uses #"..." for something else, so this ticket has been rejected.
#" is rarely used. Of my 39 test Schemes, only Racket, Gambit, Larceny, IronScheme, Shoe, and and XLisp do something other than error out as soon as they see it -- but in most of these cases, I don't know what they are actually doing.
On the other hand, Racket, Gauche, MIT, Gambit, Chicken, Bigloo, Kawa, SISC, Chibi, Chez, Ikarus/Vicare, STklos, KSi, Rep, Oaklisp, and Owl Lisp all understand |...| syntax.
Scsh, Guile, SCM, NexJ, Shoe, TinyScheme, Dream, Scheme 7, BDC, Elk, UMB, VX treat | as an ordinary identifier character.
Scheme48, Larceny, Ypsilon, Mosh, SigScheme, RScheme, XLisp, Schemik all report syntax errors.
IronScheme gets hung up reading characters from the input.