What are the technical aspects to tracing a phone call; is it more difficult for mobile phone?
In the old days, signaling was inline, hence the 2600hz hack. Calls were setup as one switch talked to another, then another, and so on until a circuit was established end-to-end. In the modern age, everything is out-of-band over SS7 and every switch is lined up at the same time. The calling station is identified at the start and no tracing is really necessary.
Mobile phones do take more effort because a mobile number isn't attached to a given switch. Thus, while the far end knows what the number is, where it is involves extra technology. The cellular phone company can identify what towers the phone is associated with and thus instantly know the region it is in. Further narrowing can be done based on signal strength comparisons, which of the tower's directional antennas are holding the signal, and GPS chips in phones.
Is it more difficult if the phone is on, but not actively being used to call?
Only a custom phone would act in a way where it didn't respond to the tower asking a question, so generally no.
Why are the tools necessary to trace phone calls not available to the general public? We have traceroute to find routing information for IPs -- why not phones? Is it a question of specialized equipment, access to telecomm systems, etc. or more social?
Social legacy and equipment access. The Internet doesn't have a separate signaling band and is based on the idea of independent operators controlling where their traffic goes. The phone company is based on the legacy of one company running the show. Switch access in the phone world is internal only to the phone company or whoever they want to specifically include. The Internet, on the other hand, doesn't really have a way of considering nodes special since everything is in the same band.
How does one prevent a (mobile) phone from being traced?
Nothing will save you from being traced down to the tower you're using, but you can really screw around with the triangulation metrics by using a directional antenna and some weak false associations or intermediary transmission layer such a radio that links you to your phone. In that case, finding your phone would leave the person chasing you still lacking a physical connection and having to trace something else. Done right, you can turn the default, "Within 100 feet," into, "Somewhere in this 20 square mile cone." That is a big time, knowledge, and equipment cost commitment, though.
You may also find some success in delaying tracing by using intermediate PBX systems to mask the actual caller. If you have dial-in access to a company's PBX, the trace will stop there and somebody will have to look at logs of associated calls into the system to try and correlate the responsible line. Nest a few of those and you may buy some time. You'll probably still eventually be traced no matter how short the call was, but it will no longer be instant.