The 1-2-3 Conjecture asks whether every graph with no connected component isomorphic to K2 can be 3-edge-weighted so that every two adjacent vertices u and v can be distinguished via the sum of their incident weights, that is the incident sums of u and v differ by at least 1. We here investigate the consequences on the 1-2-3 Conjecture of requiring a stronger distinction condition. Namely, we consider two adjacent vertices distinguished when their incident sums differ by at least 2. As a guiding line, we conjecture that every graph with no connected component isomorphic to K2 admits a 5-edge-weighting permitting to distinguish the adjacent vertices in this stronger way. We verify this conjecture for several classes of graphs, including bipartite graphs and cubic graphs. We then consider algorithmic aspects, and show that it is NP-complete to determine the smallest k such that a given bipartite graph admits such a k-edge-weighting. In contrast, we show that the same problem can be solved in polynomial time for a given tree.