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#JavaScript (ES6), 73 56 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))