Even if all weapons do the same damage in OD&D, they aren't all the same. For example, bows and swords: you can use bows from a distance. In fact, you can distinguish different missile weapons from each other based on who can attack whom at what distance, with either the hard numbers (comparing the ranges of short bow and long bow on a table) or with abstract comparison, giving the archer with a longer range the chance to move out of the shorter-range bow's reach.
The same principle can be applied to thrown weapons: smaller and more aerodynamic weapons like throwing daggers can be used at a greater distance than a sword. It can also be applied to melée weapons; a fighter should be able to maneuver out of a magic-user's reach, if the fighter is using even a short sword against a magic-user's dagger.
Instead of using miniatures or tracking exact distances, I prefer an abstract approach to combat maneuvers. If I judge that one combatant's weapon has a longer range, that combatant can attempt to maneuver out of reach, then continue attacking. My original approach to this was to give the maneuver-er a 3d6 vs. DEX roll to move out of range, although now I think I'd just have both sides roll a d6 for reaction time, lowest roll goes first, adjusted for relative DEX and movement rate. The combatant with the longer weapon can still attack while maneuvering, although perhaps at a slight penalty; on the next round, the combatant with the shorter weapon must flee or break off, throw a weapon, take some other option, or attempt to move closer to attack.