R, 35 bytes

rank(c(6,0,5,1:4*0,1:3)[scan()])[1]

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• -6 Bytes thanks to @JayCe suggestion to switch to full program instead of function

The program returns 2 for 'greater than', 1 for 'less than', 1.5 for 'equal'

Explanation :

      c(6,0,5,1:4*0,1:3)[v]          # extract the score of each card in v (got from scan());
                                     # cards in v are used as indexes in the cards rank 
                                     # vector, which is based on briscola scores vector 
                                     # c(11,0,10,0,0,0,0,2,3,4) but divided by 2 and rounded 
                                     # to integer preserving the original order

rank(                      )[1]      # rank returns : c(1,  2)   if v[1] < v[2]
                                     #                c(2,  1)   if v[1] > v[2]
                                     #                c(1.5,1.5) if v[1] == v[2]
                                     # and we select the first value