Leveling up is probably the most common mechanic for RPG's, some people might even define the genre by it. Sadly such a definition would leave out many classics including Shadowrun, Dungeon Maker, and Zelda. It would also alienate modern games like Eve Online and Blackguards. Experience points and levels are a tried and true system with widespread appeal, but I can't help but think a lot of innovation is held back by them.
Let's start with the good points of exp and levels. The most important thing it does is provide the player with a clear measure of progression, and gives them an easily discernable reward for their actions. It makes it easier to create a story driven experience in an open world setting by giving a clear indication of places the player isn't ready for. It also makes it easier to guide the players progression when things like exp curves are implemented to deter over leveling. It can serve as a way to gate content from equipment to blocking off areas in an open world while maintaining the appearance of being open. Lastly, it is an easy method to increase skills, abilities, or tie to a skill tree.
On the downside, levels are neither immersive nor organic. You can spend hours cooking in order to hit harder with a sword. Why are you unable to use level restricted gear in the first place? How does using your social skills to solve a dispute give you more health? Why does killing monsters make you better at crafting or public speaking? It doesn't feel natural, neither is it believable. You have the same equipment repeated over and over to cover gear for all the levels. What's the difference between a Bastard Sword and a Dire Bastard Sword? Well, one requires a higher level to equip, and it's dire. Then add on having to split skill points between combat abilities and utility skills like crafting and speaking and the whole thing becomes quite restrictive.
Progression and Immersion
Systems without levels are not automatically more immersive. Putting exp into a generic pool to be used on anything runs into the same problems as leveling up by cooking. Equipment can still be locked behind arbitrary skill requirements. Then there are RPG's that used the level system but found other ways to show specialized progression. Skyrim is a game, I'd argue, would have been better without levels. To increase your skill with magical arts you had to use magic, to increase your skill with weapons, you had to use them. Another hybrid to consider are the Rune Factory games. Rune Factory had skills for everything, eating, sleeping, bathing, socializing, if it happened in the game there was a skill for it. Skills increased by performing relevant actions, swinging a weapon increased the effectiveness of that weapon type, getting hit increased health, getting hit with lightning increased lightning resistance. Rune Factory might have taken it a little too far (sleeping skill?), but when the benefits from all the skills are added up, they easily overshadow the gains from levels and served as the primary means of progression. While it makes sense that swinging a hammer at an anvil will make you a little stronger, it is good for a progression system to reflect the player's actions.
Another thing to consider is decreasing the emphasis on stat increases. I'm not saying to get rid of them altogether, it's an important facet of progression, but minimizing their importance can lead to more creativity and innovation. When Soul Calibur 2 introduced its equipment system, very few weapons altered the damage done. Usually they increased the reach or how hard an opponent was knocked back. Some modifiers got a little crazy by pulling opponents toward the player or making the weapon invisible. Sometimes the focus of progression relies so heavily on stat increases that developers become blind to more interesting ideas. A lightning bolt doing a little more damage is just more of the same, but when it bounces to other targets, stuns, or magnetizes the target causing nearby metal objects to strike them, the improvement has a lot more impact. Balancing numbers is easy, coming up with new twists to old tricks is a lot more work, but also a lot more satisfying.
It is a shame so few games venture outside of the traditional level and stat systems to push the boundaries and offer a more dynamic experience. Realism and immersion aren't necessary to make a good game, in fact too much focus on them can ruin an otherwise great experience. I'm not saying level and stat systems need thrown out, they're popular for a reason, but developers shouldn't be afraid to innovate and look for other means of progression, especially when attempting to push the boundaries of immersion.