Anyway, I think that such a comparison might be of interest to several people out there. This is meant to be a a comparison, not an invitation for "edition waring". Any comments going down that road will be deleted.
Both games are based on early editions of D&D. 13th Age via OGL, D&D 5e by virtue of being the 5th edition of the world's most popular role-playing game. 13th Age brings a lot of awesome to the table that's partially derived from "indie" game mechanics, but also can be ported to other games derived from D&D.
- Ability Checks: They usually are made using the attribute modifier + level + applicable background.
- Armor Class & Damage: Are hard-coded into your class depending on the type of armor and weapon you use. Damage also scales with level (e.g. longsword wielded by a level 1 character does 1d8 damage, wielded by a level 5 character it causes 5d8 damage).
- Adventuring day: For purposes of "daily" powers, an adventuring day consists of four encounters. No matter how much in-game time elapses between the encounters. After that time, the characters also fully recover their hit points and recoveries.
- Background: A background is basically a replacement for skills. You can be a Soldier +3, Hermit +5, and the rulebook encourages you to use colorful and evocative names for them.
- Classes: Barbarian, Bard, Cleric, Fighter, Paladin, Ranger, Rogue, Sorcerer and Wizard in the 13th Age rulebook, and Chaos Mage, Commander, Druid, Monk, Necromancer and Occultist in 13 True Ways.
- Class Abilities: They are structured somewhat like in D&D 4e into at-will, daily and encounter powers. They are short and complete rules snippets that determine ability used to hit, effect and damage.
- Combat: Doesn't use a grid, does use abstract distances.
- Default Game Setting: The Dragon Empire in an area about as large as the eastern Mediterranean and environs, centered around the Midland sea. The setting is a bit quirky and leaves much room for the GM to add stuff to.
- Defenses: Instead of saving throws, 13th Age uses Physical and Mental Defense as static values.
- Escalation Die: After the first round of combat, the d6 used as escalation die starts to increase by 1 each turn of combat. You get to add the die to your attack rolls making it more likely to hit a monster in later rounds. Also some nifty effects are tied to the escalation die, both for monsters and characters. (Note: in D&D 5e that die isn't really necessary, because as of this writing AC seems to be a lot lower than in D&D 3e or 4e).
- Feats: Gained at every level. They give a small advantage and are directly tied to class abilities (some are general feats).
- Hit Points: You generally start out tougher (about three time as tough as in D&D 5e), but monsters do more damage.
- Icon Relationships: You initially get three points to determine your relationship to iconic NPCs and their organizations in the game world. Relationships can be positive, negative or conflicted. These relationships will affect game play via relationship rolls on d6s, where 6 means a good result and 5 means a mixed bag.
- Levels: 13th Age game play stretches across 10 levels, divided into 3 tiers of play that determine how difficult things like skill checks are, how much damage you take from traps etc.
- Races: Dwarves, Elves (High, Wood & Dark), Gnomes, Halflings, Half-Elves, Half-Orcs as major races, and Aasimar, Dragonspawn, Tieflings and Forgeborn as optional races.
- Recoveries: Each character has a number of recoveries that allow him to regain hit points by rolling a number of dice dependent on his level (e.g. level 1 characters roll one die, level 6 characters roll six dice). The die size depends on class. Once per battle, one recovery can be used as an action. After a battle you can also use recoveries.
- Rituals: These are spells cast for story effect and the GM will tell you how long it takes to cast the ritual and what rolls to make.
- Spells: Spells are learned in slots and come as at-will, daily and encounter variants, their effect (often damage) increases depending on the slot they are in, but as you advance your slots advance with you (i.e. you loose your lower level spell slots). Each spellcasting class gets about as many spells as other classes get class abilities.
- Two Weapon Fighting: If you carry a weapon in your off-hand, you are allowed to make an additional attack roll, if your first attack roll came up as "2".
- Ability Checks: They are usually made using the attribute modifier + proficiency bonus when applicable.
- Advantage/Disadvantage: This nifty mechanic allows you to roll 2d20. If you have advantage, you take the higher of the two results. If you have disadvantage, you take the lower of the two results. This replaces modifiers to the die roll.
- Adventuring Day: An adventuring day is tied to in-game time. You can have one long rest per adventuring day that completely recovers your hit points, and half your hit dice.
- Armor Class & Damage: Both are determined from class-independent tables.
- Backgrounds: A background provides your character with several proficiencies in skills and tools, some languages and social ability. You can be a soldier or hermit and gain proficiencies associated with that background.
- Classes: Barbarian, Bard, Cleric, Druid, Fighter, Monk, Paladin, Ranger, Rogue, Sorcerer, Warlock and Wizard all right in the Player's Handbook.
- Class Abilities: They are very different from class to class. Usually they interact with the rest of the mechanics instead of being self-contained rules snippets.
- Combat: Doesn't use a grid, but defined distances.
- Default Game Setting: The Forgotten Realms. A sprawling world, defined in literally hundreds of books and novels. But apparently, that's only part of the setting called Multiversum that contains all the D&D campaign settings and the planes, plus whatever your DM comes up with.
- Feats: Stronger game-effects than in previous editions, but completely optional. If allowed by the DM, maybe taken instead of an attribute raise (roughly every four levels, none at character creation unless you are a human).
- Hit Dice: Your character has hit dice equal to his level. Their size depends on class. You can take a short rest of one hour and then roll as many hit dice as you like to recover lost hit points.
- Hit Points: You start out with a small amount of hit points determined by class plus Constitution modifier. Each time you gain a level you can decide, if you want to gain a slightly above average of hit points or if you feel lucky and roll the hit dice. Hit points scale a lot like in earlier editions, but instead of d4 you now use d6 for the former d4-using classes. The d6-using classes moved up to d8.
- Levels: D&D 5e has currently 20 levels of game play divided into four tiers that are used to determine which adventures are suitable for your character (especially in the Adventurer's League).
- Proficiencies: Your character can be proficient in various weapons, armors, skills, tools, saving throws and languages. If you are proficient, you add your proficiency bonus to the roll. The proficiency bonus rises after every 4 levels. Armor and languages work slightly different.
- Races: Dragonborn, Dwarves (Hill & Mountain), Elves (High, Wood & Dark), Gnomes (Wood & Rock), Halflings (Light-Foot & Stout), Half-Elves, Half-Orcs and Tieflings.
- Rituals: These are spells with the ritual descriptor. They can be cast in a short ritual and don't have to be prepared in advance, if cast that way.
- Saving Throws: Instead of static defenses, D&D 5e uses six saving throws (one for each attribute).
- Spells: Spells are prepared in slots, they don't increase their effect based on your character level, but instead increase their effect based on the slot you use to cast them. The PHB will contain more than 300 spells for all spellcasting classes.
- Two Weapon Fighting: As long as you only carry a light weapon in each hand, you can attack twice with your full attack bonus.
My own conclusion: I went the D&D 5e road starting with D&D Basic. That wasn't motivated by any of the above, though. I am still torn on the issue which system to use. But when I asked my gamer friends, if the would like to play a regular 13th Age campaign, I was met with lukewarm interest. When I told them, I think about doing a D&D campaign, the reaction was markedly different. There was enthusiasm or at least a "Sure why not?" response. Therefore, D&D 5e was a no-brainer despite the excellent support from Pelgrane for their game system. D&D 5e was simply "better" at grabbing the attention of my friends.