#JavaScript (ES6), <s>73</s> <s>56</s> 54 bytes

Getting the ball rolling; more golfing to follow ...

Takes input as an array of 4 integers (`[h1,m1,h2,m2]` in 24 hour format).

 ([g,l,h,m])=>Math.min(2+h+m,h-g+24*(h<g)+m-l+60*(m<l))