I'm posting this as a companion to my thread on what Eldritch does better, over in the Eldritch Horror forum. This is of course all my personal opinion. For the sake of fairness I am trying to keep things down to the basic Arkham set.
Arkham has a much better emphasis on character than in Eldritch. While the Investigators in both games are adequately served, Arkham has a better focus on the non-player characters in the game. Ma Mathison, Doyle Jeffries, the Sheldon Gang, Deputy Dingby, - all of these characters have a real feeling and presence in Arkham. Likewise Allies are named characters taken straight from Lovecraft, where Eldritch only has generic "Hired Muscle" or "Lodge Researcher".
While the dice that come with AH aren't great - that's why they sell the neato custom dice separate - they are at least the standard white ivory tumblers, and there are five of them. Eldritch comes with four little black plastic dice that look and feel terrible, and are easily lost. Between the two dice sets, Arkham definitely wins.
While I like the countdown style in Eldritch, I don't particularly like how Doom is calculated with the complexity of counting gates and shifting the Omen marker back and forth. A lot of the time the Omen Track just seems confusing. Likewise, I like how Arkham gives a steady progress to Doom. In Eldritch it can jump around. It's not too hard in Eldritch to not have Doom advance for several turns, and then take a massive jump on a bad turn. I don't care for it.
(Because the way the rules are worded in Eldritch, it's better in game to close all but one Gate so the Doom advances more slowly. I find this a bit counterintuitive.)
Arkham, as a base set, comes with 16 Investigators. That's a lot of variety, and a lot of combinations. EH comes with 12 Investigators, and we will likely have to wait for several expansions before seeing the rest.
EH has nine city locations, roughly corresponding to Arkham's 16 stable locations.Just to start of with, Arkham has more locations, but these locations also feel more flavorful. EH encounters are more mechanically predictable, but Arkham is usually better at more narratively predictable encounters
EH has six Expedition Encounter areas, which correspond to Arkham's unstable Encounters as they are high risk, high reward areas. Such encounters are hard to get to, and very predictable. Arkham offers more flavor and more variety, even in the base set.
Other worlds in Arkham have a very definite flavor. Encounters are set, and you have a certain amount of judgement about who to send to what Otherworld. In EH Other Worlds are just a big mess. You could be sending an Investigator anywhere, and they all feel kind of same-y.
Rumors in Arkham are limited, and have a reward for completion. They make the game more challenging, but also can help provide opportunity. Rumors in Eldritch Horror just screw you over, and can end up in rumor pile-up. I have lost games because of multiple rumors popping up simultaneously, and it really isn't that rare.
Standardized Mythos Phase
I will say it really helps that 99% of Arkham Mythos cards are exactly the same format.
Gate, Clue, Movement, Effect. It's very predictable. Eldritch's varying format in Mythos can make it very easy to forget or skip over something by mistake. Half the time I forget to apply the actual card's effect after applying all the Reckoning effects.
If I think of more things, I will post them.