I'm a Marriott Platinum Premier and a Hilton Diamond. I use both programs quite a bit...they each have their strengths and weaknesses.
1 - Discounted rates. Not because of elite status, but then again I wasn't seeing many Gold/Plat discounts at Marriott either. The M12 / E0G / E0P codes of the Marriott of yesteryear haven't been useful to me in quite some time. Hilton has MVP rates that sometimes beat AAA by a little bit - just google them here. Available to the public, status not required.
2 - Late checkout at Hilton is always a used-car haggle with the front desk. I'm Diamond, can I have 2PM? "I have to speak with my manager." DYKWIA? I'm Diamond! "The boss says I can do 1PM." I wish Hilton would formalize this (even if it's just 2PM) the way Marriott/SPG did.
3 - Others have covered the lounge brands: the one thing I'll add is that when Hilton has a lounge, it's *usually* open every day of the week. There is much less weekend closure of lounges than at Marriott. There may be fewer total properties at HH that have a lounge than at Marriott, but when the HH lounge exists I find it to be a little better than what a Marriott lounge would be for that location.
4 - Breakfast at non-lounge hotels. The benefit isn't materially different than Marriott, and *most* hotels either offer a substantial buffet or enough of a credit to order two decent breakfasts from the menu. There's the occasional bad apple that has a lame continental offering, but I haven't run into too many. The only complaint I've had is that there are some Embassies out there that have a vastly area that is vastly too small for the number of guests in the hotel. The only explanation I have for this is that the property was built as something else that didn't offer breakfast for all and converted into an ES. HGI is a sweet spot in the HH lineup - good hotels with pretty good breakfast, a very nice upgrade from Courtyard IMHO.
5 - Some Embassies put real effort into their receptions with hot food, local brews/wine, etc. Others just mail it in with pretzels, Bud Lite, and box wine. Homewoods sometimes have a little more hot food but it's pretty low quality. I like both of these brands but the receptions are not a particularly high-value component. If you run into an ES serving good local beer, consider it a bonus...but don't assume either of these brands are going to serve a culinarily sophisticated offering.
6 - Diamond does not guarantee upgrades, but in my experience neither does Marriott Platinum Premier. On the upgrade front, neither are very good in most places. My best Hilton upgrades have been Middle East and Asia. That's also where I've gotten great SPG upgrades, so my guess is that the heritage-Marriott brands are similar. I consider Diamond and Plat Premier pretty equal on this front. In the U.S. and Canada, neither lead to suites for me. They're just insurance against bad rooms.
The other pro-HH thing I'll add is that they've never penalized elites at "resorts". I know Marriott is removing the resort penalty, but until I know for sure they have good offerings, I still prioritze Hilton for resort stays. Award stays at resorts are truly $0.00 - no junk fees.
HH has some partial elite recognition at W=A, a little more than Marriott has at Ritz, although I don't have any 2018 experiences. (I might have read that W=A benefits were improved recently?)
In terms of hotel quality, I've always felt that HH > MR in the low- to mid-range. MR/SPG > HH when you get into luxury/unique properties (mostly thanks to SPG).