Interesting how all the responses are pretty different so far. And mine will continue that...
The biggest similarity between the two is their affinity for big bold features that encourage heroic shots. This is most clear in their par fives, which usually are very creative.
The second, closely related to the first, is that both build courses with a strong intent on fun. Obviously they don't succeed with every golfer, but their styles are both unique and their "fans" strike me as being pretty devoted. I make an effort to see courses by both of them, so fair enough to count me in!
They both take risks in terms of severity/playability on their courses, and I would argue both cross the line at times. Despite that, for better players one could argue that both design courses that look harder than they play.
For all those similarities, their courses certainly look different and are bold in different ways. I tend to disagree that Engh isn't interested in the visuals, I just don't think Engh is interested in making visuals look natural. Engh courses have a certain look to them, which seems unlikely if the visuals weren't of some importance.
I would have a hard time picking one vs the other. Black Rock and Tobacco Road succeed in being all about fun, IMO. Their lesser designs are still fun IMO, even if Red Hawk Ridge (Engh) is blander than his other work and Tot Hill Farm (Strantz) is a bit odd in places.