#JavaScript (ES6), 73 56 54 52 bytes
Uses 24 hour format. Takes input as 4 integers, where g & l are, respectively, the hours & minutes of the current time and h & m are the hours &minutes of the target time.
(g,l,h,m)=>Math.min(2+h+m,h-g+24*(h<g)+m-l+60*(m<l))