Generally speaking, Shot pattern density. A 1 1/4 ounce load of shot is likely to produce a denser pattern than a 1 ounce load of equal shot size.
This usually translates into the ability to put more pellets on a target, such as ducks, at a greater distance than the lighter weight load.
More drams of powder are also used to propel the heavier load at an adequate velocity.
The down side off denser pattern is that closer shots often lead to mutilations of the target or misses. More shot and powder also means more recoil.
For example - one ounce of shot (for example size 6) may contain 400 pellets. Anything more than an once will contain more, say and once and a quarter of size 6 shot has 470. Those extra 70 pellets fill in the gaps left in your shot pattern making it more likely one strikes a vital spot on your quarry (more dense of a pattern as mentioned above).
A draw back to having a lot of shot is if you hit a small game animal at close range with a lot of pellets its meat will be ruined.
Seeing your handle, "duck hunter", if you may be asking this question in reference to steel shot, the answer may be a little different. I know personally when waterfowl hunting with steel, I think it's really basic, other than you need to inject speed of the shot into the equation. Many waterfowlers feel that when using steel shot, speed of your shot is much more important than the load size. Since late season waterfowl are pretty thick with feathers and down, and since steel doesn't have the penetrating effect of lead, you can make up some of that disadvantage by increasing the speed of the pellets. Maybe giving a better penetration factor?
I believe it's a choice you have to make, either more pellets, or more speed. Some 1 1/4oz steel loads have a speed of over 1600fps, but step up to a 1 3/8 load, and your speed goes down to about 1340. I believe that is why 3 1/2" loads are becoming more popular. I normally never shot 3 1/2" shells, simply because they weren't really needed, but the 3 1/2" now offers you the best of both worlds. My shell for a normal late season duck or goose hunt is a 3 1/2" federal steel load, it's 1 1/2oz of shot, and is about 1450fps. So i'm getting a good amount of shot, but still have a good speed as well. Now you hunting situation should also tell you what to shoot, what I do, is try and find a shell that is the same speed, but maybe go back to a 3" shell or 2 3/4 shell when hunting early ducks, or close decoying ducks, if you know what i'm saying.
I'm no expert, and it probably doesn't make much difference, but common thinking to me that would be if you change the speed of your shell, I would think your swing and lead would change as well. That's why I try and stick with the same speed shell, you can compensate by using a smaller shell, or smaller shot, and not ruin the meat of early or smaller ducks, as the earlier post said. I feel sticking with a similar speed shell, it keeps my shooting more consistent, but that could really all be in my head, but that's what shooting is all about.
I just thought with his title, he might have been talking about waterfowl, that's all. I know very little about the lead stuff, except it's relatively slow compared to todays steel stuff, I don't think I've ever shot any lead, except for a turkey load here and there, and trap loads. I would have no idea what I'm talking about with lead loads for upland or small game hunting. I do know with talking to the "old timers", 3", #2 copper plated lead was the ticket for ducks, way better than what we have in plain steel choices to day....