QuasiWizard

In games (in particular RPGs), hierarchical advancement (a term I came up with) is progression towards a known goal. It often includes progress markers, and gives boons that may not be directly related to the experience of the adventure. The most common example is levels. It proceeds in one direction, towards higher levels, and often gives mechanical bonuses along the way that didn't come from a super specific experience in the adventuring process. It is instead a marker of having gotten better over time, awarded at the level-up. You don't need explicit levels to have hierarchical advancement; every time a number or feature becomes unarguably better, it advances hierarchically. In its most basic form, it is "I get better."

Lateral advancement (a term I came up with as well) is "advancement" where there is no solid plan or guarantee of an improvement; instead, characters change in often incomparable ways through the specifics of an adventuring process. These changes are not necessarily a unilateral boon, but instead give choices and tools. This is very related to diegetic advancement, and is likely its most common form, but is not identical. You can still proceed hierarchically and diegetically. Just make the next tier of advancement reified in the world and obtainable. However, most that play with diegetic advancement implement it laterally: there might be a combinations of benefits at a cost, or may be too specific to be worthwhile to fill an inventory slot. In a nutshell, it is "I have changed."

These ideas play into incentive in RPGs. With a hierarchical XP system, players know the way to advance, and what to look forward to. It can direct a session and make it easy to follow. It can rub some players the wrong way though, as it can occlude other actions that might warrant reward, but the hierarchy doesn't award anything. Lateral advancement is much more freeing in this regard; you have the freedom to award something more free-form to the character. That's not to say lateral is better. I have had some players get annoyed at the lack of foresight, with no idea what their player will become without the known hierarchy. Neither is better, they just appeal to different gamers.

Advancement is also relevant to game crunch. Though only correlated, more hierarchical advancement is included in crunchier games, and more lateral advancement in rules-light games. These are naturally included by the simple amount of content: crunchier games have more content, and thus will lay out many advancement paths to give the desired crunch. Rules-light games give less direction on characters' advancement, and gives the GM a chance to affect the characters and let their choices shape them. Again, neither is better, they just appeal to different gamers.

These are not mutually exclusive, either. OD&D, for instance, has hierarchical advancement. Most games of OD&D will include lateral advancement as well, in the form of items or effects. They can be freely mixed, often producing interesting combinations.

I don't have a conclusive ending to this post. Take these concepts, and accept, reject, or puzzle over them. Just don't design for either type of advancement, design for your desired world.