Now, before I get into this, I want to say that I like games with both. However, I was wondering what other people prefer.
Some games, like Castlevania Rondo of Blood, Lament of Innocence, the Legend of Zelda, and Darksiders (to name but a few) use the item progression method to increase a person's power and ability to move forward. This system is very rewarding, when you find heart container peices, or things like the hookshot to further your quest there is a real sense of accomplishment. It also lends itself to things like power upgrades, In DMC or Darksiders you can buy new weapons and skills from the store or upgrade your combos (a la Lords of Shadow). This allows for several variations of the same character from game to game, it also allows for different combat combinations depending on your play preference. For instace in DMC 3, maybe you prefer the guantlets and greaves for the kicks and punches, or perhaps you like the electric guitar for it's ranged electrical attacks and ability to become a scythe. Each of these weapons has specials and combos a person can unlock with whatever currency the game requires (red souls in DMC 3, or human souls in Darksiders for instance). Some issues that could be raised with this system are it's reliability on linear gameplay, you require these items to move forward. You also have to remember where all of the areas that you've seen that require items you currently don't have are, or else you won't collect everything. This creates a lot of backtracking for the player, don't get me wrong the exploration increases the replayability and in the case of Castlevania games it is essential to get full completion.
Others, like Final Fantasy, and some Castlevania games (SoTN, Aria and Dawn, or Order of Ecclesia) use the level up feature. Now, before we get into this I realize they also use items, such as weapons or armor, however, these items are not always required to move on. In most modern RPGs, weapons are the key items that you will use. For instance, Mass Effect placesa lot of emphasis on the guns and armor you use in combat, as well as the loadouts of your squad mates. In games like Final Fantasy Tactics the job's you choose require specific sets of armor but you gain exp. for both your standard character levels, and your job levels to gain abilities for thatjob. In many games this is a great device to enable your characters growth, and item advancement is difficult for a party of characters. However, this also makes some games too easy as you can grind and Dragonball Z your way to victory. Can't beat ruby dragon in the Nibelheim reactor? Grind five or ten levels per character and take another shot at it. Some games use both though, and they're excellent uses of items that level up as you do. The World Ends With You uses pins as weapons, however, you need to both collect them and level them up to advance throughout checkpoints and defeat stronger enemies. It still retains the character leveling, and exp notions though.
The last thing I should probably mention before I go is that some of the games I mentioned on the leveling up side still use heavy item progression. Castlevania has always used relics in the games where leveling up exists, you require these relics to move on. Also, some types of item progression game use leveling up as well. Monster Hunter uses tempering and forgeing to create armors with different defense and offensive ratings as well as immunities to different elements, and demon's souls allows you to level up weapons and armor using souls and materials.