Through the help of karlkoeller and Herbert I came upon the following solution,
\documentclass[10pt]{article}
\usepackage{mathspec}
\setmathsfont(Digits,Latin)[Uppercase=Italic,Lowercase=Italic]{Helvetica}
\setmathsfont(Greek)[Uppercase=Regular,Lowercase=Italic]{Symbol}
\setmainfont{Helvetica}
\usepackage[italic]{mathastext}
\begin{document}
\Huge This looks right: $\delta_e/L_e$ (\%)
\Huge This looks wrong: $\delta_e/L_e\,(\%)$
\end{document}
which produces,

I am relatively certain I tried this back when karlkoeller suggested mathastext on Dec 28th, and I ran into a bunch of errors. In between Dec 28th and Jan 1st, I updated from TexLive 2012 to TexLive 2013, and now it appears to work. The mathspec package was not updated between 2012 and 2013, but the mathastext package was updated something like 15 times. Probably one of those updates fixed the errors I was running into.
In the interest of informing others, I thought I would also share an alternate solution that I think I like better. The XeLaTeX package is nice, but it takes nearly 3 times as long to typeset a short document like the one above. Also I am not sure the Symbol and Helvetica fonts will work so well when typesetting complicated mathematica formulae. Instead I have settled on the newtx package, which has full math support and a Helvetica clone for its sans-serif font. The following code
\documentclass[10pt]{article}
\usepackage{newtxtext}
\renewcommand*\familydefault{\sfdefault}
\usepackage[italic]{mathastext}
\usepackage{newtxmath}
\begin{document}
\Huge This looks right: $\delta_e/L_e$ (\%)
\Huge This looks wrong: $\delta_e/L_e\,(\%)$
\end{document}
produces,

The \delta is different in newtx, but the overall style is similar and I was not married to the Symbol font. I just did not want the computer modern (CM) fonts. (IMHO the CM fonts line weights are too light.) Unfortunately, close inspection reveals that the newtx parentheses are a bit different in math mode versus text mode, but I can handle that.
mathspechas known problems with those kind of symbols. You might have a look at themathastextpackage. – karlkoeller Dec 28 '13 at 11:51\usepackage{unicode-math} \setmathfont{Helvetica}in the preamble – karlkoeller Dec 29 '13 at 07:56