Greater responsibility should equate to greater pay.
Not always. I'm pretty high up in my firm and of the 100 or so people below me there's a fair few who get paid more than me despite my having way more responsibility. Generally the market rate for a given role is the biggest influencing factor in pay.
Having said that, at the employment end of the spectrum the OP is at the choice is pretty clear. Politely talk to your boss about a pay increase. Forget about the other person and explain why you do a great job and are deserving of it. Or secondly go and find another better paid job.