Fourth Edition does many things in this vein. Certain weapons had better chances to hit, certain did more damage, certain had higher minimum damage.
But more than this, Fourth Edition has many feats, styles and powers specifically designed to reinforce the strength of each weapon. There are even paragon paths built around specific weapons.
However, I do agree that Fifth Edition should refine those ideas (not all of them worked well).
I think one of the biggest issues for certain weapons and armour is the economy in 4th edition: beginning characters can buy plate armour and a two-hander, so no one is ever so poor that he need buy a club. But I think the solution there is simply to add optional rules for zero level characters (or as someone pointed out an entire five level sub-tier: most games would start at Heroic Tier, but some could start at Adventurer Tier with card tricks instead of cantrips, quilted armour and rusty farm tools.)