Customizing latexdiff default (UNDERLINE) output styling

There is no standard option to do this, but you can create your own preamble file for latexdiff to replace the highlighting commands. All you need to do after a standard run of latexdiff is to copy the lines latexdiff writes in to the header of the diff file and change those that do not fit your style. In your case the command to change is \DIFadd that by default is

\providecommand{\DIFadd}[1]{{\protect\color{blue}\uwave{#1}}} %DIF PREAMBLE

The wavy underlining is generated by \uwave{...}, so removing this we get

\providecommand{\DIFadd}[1]{{\protect\color{blue}#1}}

A complete preamble file is then

\RequirePackage[normalem]{ulem}
\RequirePackage{color}\definecolor{RED}{rgb}{1,0,0}\definecolor{BLUE}{rgb}{0,0,1}
\providecommand{\DIFadd}[1]{{\protect\color{blue}#1}}
\providecommand{\DIFdel}[1]{{\protect\color{red}\sout{#1}}}                    
\providecommand{\DIFaddbegin}{}
\providecommand{\DIFaddend}{}
\providecommand{\DIFdelbegin}{}
\providecommand{\DIFdelend}{}
\providecommand{\DIFaddFL}[1]{\DIFadd{#1}}
\providecommand{\DIFdelFL}[1]{\DIFdel{#1}}
\providecommand{\DIFaddbeginFL}{}
\providecommand{\DIFaddendFL}{}
\providecommand{\DIFdelbeginFL}{}
\providecommand{\DIFdelendFL}{} 

which if you call this preamble.tex can be used by latexdiff via

latexdiff -p preamble.tex old.tex new.tex > diff.tex

With new.tex

\documentclass{article}

\begin{document}

The text now starts with this.

\end{document}

and old.tex

\documentclass{article}

\begin{document}

The text started with this.

\end{document}

this generates a diff.tex that produces