Do résumés written in LaTeX show more promise when applying to graduate schools? I am a computer science student from India. I noticed that most of the résumés and thesis papers etc. I see are written in LaTeX. 

If I write my résumé in Word (say) rather than LaTeX, will it have a negative affect when I apply to graduate schools in the U.S.? 

If not, then should I even learn LaTeX for my future academic prospects?
No. Almost no one cares. You should learn LaTeX if you intend to work mathematics and will need to write up your work. It's much simpler to typeset formulas in LaTeX than MS Word, and it's also free.