One of the more in-your-face improvements is in computer power/reliability. Apollo had something like a 8080 in computer power(1970s-era computer). Today, you can buy COTS from XMOS development/experimental kits with well over 200x the processing power of a 8080 for $99US. Note that the COTS kits can 'move' more information, require less wattage, store more information, and have more capabilities.

Note, however, that the COTS kits are not radiation-hardened, vibration/heat-tested, and any other of the important qualifications needed for spaceflight-ready computing power. It's not a strictly fair comparison. Here's a better comparison: it's analogous to having over 500x the power. http://www.parvus.com/product/overview.aspx?prod=DuraCOR820. According to a press release, the Parvus above with off-the-shelf options is about 10K. http://www.reuters.com/article/pressRelease/idUS196111+10-Feb-2009+MW20090210

Here's an article that goes over it a little bit: http://www.downloadsquad.com/2009/07/20/how-powerful-was-the-apollo-11-computer/ It has one or two minor errors, but nothing significant.

Since the early '70s, we've had the VLSI revolution in chip design, plus there is a huge focus today on power-efficient computing. I also believe we have better rad-hard silicon composites for the chips themselves, but I won't swear to it.

On the list should be the development of a vibrant amateur rocketry community. SpaceX's main engine designer, for instance, was a member of the Reaction Research Society experimenting with his own bi-propellant engine. A lot of good experimentation is going at that level. Testing rockets is expensive, and some of the amateur results and experience has no doubt percolated up to the larger-scale operations.

A big advancement has been the use of computers in the design vehicles, ie, CAD/CAM. You can pretty much design an entire spacecraft from the ground up, test it — vibrational testing, electronics testing, computational fluid dynamics etc. — and make appropriate changes all before a single piece of hardware has been manufactured. And then the actual manufacturing is also computer aided (CNC).

Of course it's not perfect and the simulations can be wrong, but it's a pretty significant step. It's allowed for relatively cheap and rapid development, especially for startups for Armadillo, Masten, SpaceX et al.