icloud mail: "get new mail" vs. "synchronize"?

what happens in each case?

Mail synchronises with iCloud and other email services at regular intervals depending on what you have set: Mail > Preferences > General > Check for new messages. The Automatic settings is set to push it from iCloud.

You can override this by asking Mail to check for mail and it will do it immediately, regardless of the interval since the last check.

'Get Mail' is useful if you know an email has been sent and you need to get it immediately, if not sooner...but if Mail is set to Automatic retrieval it will arrive fairly quickly anyway.

If you have set the Check for new messages option to Manually you will have to click on Get Mail to retrieve it because with this setting Mail has been set not to fetch your mail by itself.

I interpret "automatically" as Push, but under a different title; and you don't get a choice of individual settings for the multiple email addresses most of us have. Like you, I found it hard to get a written set of instructions for Mail on a Mac, though I've been using it for nigh on 20 years (since OS X 10.2) so I can claim to be familiar with it.

What I find interesting in iPhone is that the settings for mail (Settings > Mail > Accounts > Fetch New Data) are a bit more variable. The options are the same - Auto, Hourly, 30 mins, 15 mins, but there is another choice to make called Push which overrides the settings you can make individually for each of your mail accounts. A case of when Push comes to shove, as it were.

I have Push off because it's not necessary and it's not as if the world needs to contact me that urgently, but mostly because over time it consumes battery power unnecessarily.

So we have on iPhone:

Push, then if Push is inactive a choice in each of the mail accounts to Fetch (ie. pull) or Manual. The latter is a fresh launch of the app or a pull-down of the items in the In Box - you will see a wheel turning furiously fetching your mail, then -

Fetch (below these choices) for mail accounts where Push is off and it's here you choose the regularity.

I have my iPhone iCloud mail set to Push (from iCloud) but my other accounts only to Fetch manually, which occurs when I launch the Mail app.

Thank you : )

...kind, and down-to-earth

I'm in Australia. It's what we do here, except when playing cricket with the English where we take no prisoners.