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By finding all integral points on certain elliptic and hyperelliptic curveswe completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for$-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\;(3,4),\; (3,6),\; (4,6), \; (4,8)\}.$ Moreover, we present some otherobservations of computational and theoretical nature concerning the titleequation.

On the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$