The Explore Difficulty screen says what amplifications of gold, mythstones, and tokens you will get.
Every battle in Explore always gets you a traitstone.
The traitstone drop rates for non-boss battles are not officially posted, but, from information I can gather, the rates don't change with difficulty, and are: 60% minor, 30% major, 8% runic, 1.5% arcane, 0.5% celestial.
The rate of Arcane Traitstones only are also increased at higher difficulty levels for the Mini-Boss battle. In particular, at difficulty 11 and 12, Mini-Boss battles give Arcanes with 100% chance. (Unofficial drop rates for Mini-Boss battles, from Difficulty 1 through Difficulty 12: 7%, 9%, 20%, 25%, 30%, 41%, 51%, 61%, 71%, 81%, 100%, 100%.)
(For people curious about token rarities and quantities, I've put together a spreadsheet for that information. One of the guild leaders I work with requested that information, so I thought it might help others too. Again, 100% unofficial.)
The Mythic Boss battle always gives an Arcane (no matter what difficulty). But higher difficulty explores (which give more mythstones) help there too, since you need to collect 100 mythstones to unlock a Mythic Boss battle.
Note that you can hoard mythstones: just because you unlock the Mythic Boss battle for a given explore run, does not mean you have to spend it! Just do an explore run in a different kingdom (especially ones with mythstone bonuses) and hoard up mythstones for doing the Mythic Boss battles for the kingdoms you want.
For example, say you are trying to get arcanes from Whitehelm. You could do all the mythstone-bonus kingdoms explore runs at difficulty 12 (to maximise the mythstones you get), without doing the Mythic Boss battle in any of them, then go and do several Whitehelm runs at a lower difficulty level if you wish.
(Why? Mythic Boss battles are all-or-nothing: if you lose, you lose the mythstones spent to unlock that battle. So unless you are confident of a difficulty 12 win, a lower difficulty level is more likely to be fruitful.)