Which do you prefer?
If you’re talking about a Chicago beef, I’d probably rather have a hamburger.
Define “a hot beef”…
Salisbury Steak, which I guess leans closer to hot hamburger.
Oh dear god, neither.
When a sandwich needs a fork and knife, it kind of defeats the purpose of it being a sandwich.
If I had the time to sit down I’d do the hot beef but I only see them open faced arond here, never with bread on top and that’s how I’d prefer it (open faced I mean).
Italian hot beef
I went with the hot beef, as it’s closest to what I eat: an open-faced sandwich with non-hamburger beef.
Anything with beef, ground or otherwise, is okay with me! 
Based on those pictures I’d go with the beef. If I made them, the beef would be piled high with rare soft as butter roast beef on a bulky roll with the gravy on the side. The burger would be 6-8 ounces of charcoal grilled ground chuck on a bulky roll, without gravy. Then I’d eat them both.
Well, the Salisbury steak has gravy, though. Yeah, it’s made of ground beef, but I think that the gravy puts it closer to the hot beef.
I’d rather have a Salisbury steak or a hamburger than a hot beef. I’d actually rather have roast beef and mashed potatoes and brown gravy. I’m really not very fond of sandwiches with gravy on them.
Based on the pictures… your “Hot Beef” is known as a “Beef Manhattan” in the Midwest.
The “Hot Hamburger” I would assume someone ran out of roast beef for their Manhattan.
I don’t really care for beef gravy. Neither. I’d prefer a regular Italian beef sandwich or a hamburger without the gravy, thank you.
I’ve seen that as a ‘hot roast beef sandwich’. For that style of ‘sandwich’, I prefer the open-face hot turkey sandwich. (Loved 'em when I was a wee un.)
If I ordered a Manhattan and got either of those I’d want a stiff drink!
Ick to both. Too much gravy.
Neither presented that way. In a showdown between an Italian Beef and a goof burger…I’ll get back to you.
But those things are abominations…gravy?
Gravy is not supposed to be soup.