The differences between the liver and muscle in the glucose utilization from blood are due to the different regulatory propeies of:

Glucokinase is an isozyme of hexokinase in the liver. It has a different affinity to glucose and different regulatory propeies. Muscle hexokinase is inhibited by its product, glucose-6-phosphate, while glucokinase in the liver is inhibited by fructose-6-phosphate. In the last case, the inhibition involves an additional regulator protein. Fructose-1-phosphate (direct product of the phosphorylation of fructose) can replace fructose-6-phosphate competitively and thus block its inhibitive influence. This explains why high levels of fructose in the blood stimulate glucose uptake by the liver. Ref: Bender D.A., Mayes P.A. (2011). Chapter 20. Gluconeogenesis & the Control of Blood Glucose. In D.A. Bender, K.M. Botham, P.A. Weil, P.J. Kennelly, R.K. Murray, V.W. Rodwell (Eds), Harper's Illustrated Biochemistry, 29e.