Firstly lets look at things such as manned flight, radio transmissions and other innovations in the late 19th century that are now critical today. These eureka moments in science and engineering were traditionally brought about by a few people and their relative complexity is low. Look at the innovations we're seeing in physics today and you're talking about teams of people across multiple generations attempting to solve problems our ancestors couldn't even imagine. Electricity producing fusion reactors like ITER (and later DEMO)? Not a chance. They were too busy trying to get manned flight to work. Breaking the speed of sound in air? No way. Putting men on the moon? Not a chance.
Here's the thing, we (as a planet) could put men on the moon. Granted we're not as ready as the USA in 1969, but given the need it could be achieved. Putting a man on mars? Well that's a lot further away much further than the moon is with a whole new raft of unsolved problems.
I can immediately think of two areas where growth is as great and immediate in the field now as it was in physics in the early 20th century - nanotechnology and biotechnology. If you look at how much of Physics we understand, and our understanding of what we can and can't manipulate versus biology you can see that there's plenty of space for innovation.
I would also make a case for disruptive versus iterative innovation. The automobile has come a long way in 100 years but is essentially the same thing, a horseless carriage that takes the driver and passengers from point A to point B. We have however had massive amounts of iterative innovation in speed, fuel management, safety and so on. The same with air travel with demands and needs changing and planes changing to accomodate. Sure, it may look like a plane but the insides of an A380 are very different to a DC-3.
If the 20th century was the century of physics, I'd say the 21st is the century of nanochemistry and biomechanics. I suspect (although have no real proof) that we don't see the changes on the same scale because our quality of life, education and so on has been brought up by such a level already that improvements have less of a social impact in the west. There's no need for a cure for smallpox anymore. Disabilities from Polio have been down in the west for decades. Diseases like SARS and H5N1 that could've wiped us out in the early 20th century like the Spanish Flu have so far been contained.