What northwind said, it's all about the trail. The distance between the wheels contact patch and an imaginary line drawn down from the headtube.
By manipulating head angle, wheel size and fork offset you can make 2 different bikes feel similar. There's a bit more to it than that 2D reprisentation (otherwise there's all sorts of angles/wheels/offsets which give the same trail figure), as you turn the bars they change, which is why wheel sizes have matched fork offsets and designers only have one remaining variable to contend with, the head angle. Or you can f*** it up, which is why some bikes feel like they fall over into corners then take a lot of effort to get back upright again. Thankfully bad bikes are rare these days.
[edit, took a while typing that]
Northwind has it, more trail for a given angle and it's the stability created by the trail that you feel.
On an only partially related note, It always surprises me that people reckon a 1 degree or even 0.5 degree steeper head angle makes a bike feel scary twitchy, I don't know if I'm doing it right or wrong but I'd have to pay close attention to feel 1 degree difference and it certainly wouldn't make me perceive the bike as difficult to ride.
I recently put an external headset on my El-Mariachi, so the front end is ~15mm taller, ~1 degree slacker, and the BB up by ~5mm. The bike feels far slacker/stable than it's old nimble self.
But that's just one variable, if you swaped between bikes telling the head angles would be harder as there would be other factors influencing the handling.