Is shutting down and restarting different from doing restart? A first-level Applecare rep recommended shutting down and then turning on my computer to deal with an issue.  Specifically, she told me to do select "Shut Down" from the Apple menu, wait 20 seconds, and then turn it back on, explaining this is different from selecting "Restart".  

This surprised me, because I thought restart incorporated a full shutdown, followed by an auto-restart.   She said no, that the software shutdown executed when one selects "Restart" is not as complete as the software shutdown executed when one selects "Shut Down".  She further explained, colloquially, that the latter allows the system to more fully reset (and that to truly get a full reset, you additionally wait ~20 seconds before turning it back on). 

Is what she is saying correct?

Independent of (but related to) this, my computer freezes unless I do a restart about once a week.  An Applecare engineer explain this is likely because of errors that can accumulate in non-ECC RAM, which a shutdown flushes out.  He said the rate at which they accumulate varies with usage.  For some reason my usage causes fairly rapid accumulation of these errors, which is why I have to reboot more frequently than most.  I have found that restart (as opposed to shutdown + restart) is sufficient for this, suggesting that the former is sufficient to flush RAM.   Is this correct?
The only difference is power is actually (mostly) removed from the components (logic board, PCIe bus, etc.)

The difference from back in the day, was called a warm restart or a cold restart .

Why the difference?

It depends on the issue.  If you're having a hardware issue like a track pad not responding or a GPU acting weird, you may be asked to shutdown and then restart, usually removing the power cord and waiting for a period of time in between.  If you're having an issue with a service, like your web server not working, a warm restart is sufficient.

So, from a software perspective, they are identical.  From a hardware perspective, the difference is power is actually removed and reapplied.

Now, usually cold (re)boots are done.

It's important to note that an SMC reset sits in between a warm and a cold boot.  When you reset the SMC, you are "resetting" the values the chip has accumulated back to factory defaults.  You're forcing the SMC to go back to when it was first initiated and needs to "relearn" everything power, temp and fan related with your Mac.  When you reset the SMC, you actually do a warm reset as well.


  ...my computer freezes unless I do a restart about once a week. 


You have a hardware issue - it's not normal to need to continually reboot a Mac in this manner.


  She further explained...the latter allows the system to more fully reset...you additionally wait ~20 seconds before turning it back on). 


To truly get a cold reboot/reset, you need to disconnect all power.  So depending on your Mac model, you'd need to disconnect the battery as well.


  An Applecare engineer explain this is likely because of errors that can accumulate in non-ECC RAM, which a shutdown flushes out. 


This statement from the Apple engineer is incorrect.

non-ECC memory isn't "building up" a cache of errors that must be flushed out;  that's not how this works.  While non-ECC memory won't "fix" the corrupted data, corruption is actually very rare, especially in desktop computers.    The worst that will happen is a glitch or a hung app.  If you're having this all the time, you may have faulty memory.

Start by running Apple Diagnostics (Hold D while booting).