Shutdown means the computer is completely off and the state of the operating system and programs are not saved. So, as you've noticed, you have to reboot and then restart all your programs, when you power up after shutdown.
Uh, "completely off" isn't usually really true. There is a bit of "standby power" coming out of the power supply. This enables the power controller on the motherboard. (The actual power your machine uses does not go through that flimsy little button on your front panel!) It also enables peripherals like USB keyboards that have power switches.
This is no different than a TV or other gadget with a remote control - when it's "off" there still has to be a little bit of power used to power the receiver for the remote control.
But after "shutdown" the computer is as "off" as it ever usually gets. I'm going to call this "normal off" below.
For a desktop or tower machine you can remove ALL power only by pulling the power plug, or flipping the hard power switch that some machines have on the back of the power supply. For a laptop you'd need to disconnect from AC power *and* remove the battery for it not to consume standby power.
I'm going to call this "really remove all power" in the next bits, so this is more important than it might seem!
Ok... in "standby", as you've noticed, the operating system and programs' state are saved. When you "resume" (as it's called) from standby (and, depending on how your machine is configured, log back in) you can get right back to work. The OS has not shutdown and restarted, nor have your programs. In the meantime, the machine saved its state in RAM. RAM does take some power to maintain.
Important: This power to maintain the RAM contents comes from that "standby" power I mentioned above. Standby state uses a little more power than does "normal off". The extra power is needed to keep the RAM contents "refreshed."
This means: if you put it in standby (technically this is called S1, S2, or S3 state), do not "really remove all power". i.e. do not unplug the desktop from the wall. Or if a laptop, do not remove AC power *and* remove the battery at the same time. If you do the OS will have to restart, and so will all the apps you had open. You will lose any work that was not saved or auto-saved, you will lose. You may even corrupt the hard drive and so have to reinstall everything.
In "hibernate" (S4 state), the system's state is saved not in RAM but in a special file on the hard drive. The machine then goes to the same power consumption as it does during "normal off." More: You can "really remove all power", let the machine sit for days or even weeks, plug it back in, power it up and it will be right back where it was when you left it (desktop, apps open, etc.).
This difference is very important on small laptops with small batteries. In standby, when not connected to AC, some power is constantly being used from the battery to keep the RAM contents refreshed. And it can't last forever. I have one laptop that can only maintain standby power to RAM for about eight hours.
If you have hibernate enabled, then if you put the machine in standby... then later when battery power is reaching "critical", the OS should wake the machine up, write RAM contents out to the hibernate file, and then go to hibernate state, all without you knowing about it.
Vista does this better: It writes the hibernate file (if enabled) when you first put the machine in standby, then puts the machine in "normal off" mode. This means much less power is needed at "battery critical" time to change to hibernate mode.
I do this stuff for part of my living.