When should I "send mail by queue" with the SMTP module? On the SMTP module configuration page at /admin/config/system/smtp, it offers the option to Send mail by queue (Mails will be sent by drupal queue api.).  

When would I want to check this box?  What are the advantages and disadvantages?
TL;DR: send the email by queue if you are sending larger number of emails at once, or if your SMTP server takes longer time to process a request (for whatever reason).
Below is a brief overview of the main pros/cons.
Pros

Better handling of large volume of emails. Email sending task will be split into batches, and Drupal will process them in smaller chunks during cron runs. This is better because your server might time out or become unresponsive for visitors if you are trying to send 500 emails at once.
If the same task is split into multiple batches, it will be easier to process them, and regular visitors will not experience as big decrease in performance.

Cons

Requires more frequent cron (or a custom script that will process the batches), which translates into more used resources at all times. You might see certain decrease in performance if the site is processing email batches and there are a lot of users online simultaneously. This of course depends on many factors (number of emails you are sending, frequency of your cron, speed at which remote SMTP server processes the requests, number of visitors on your site, complexity, and so on).
See Elysia Cron module for fine tuning the cron implementations of all modules on your site. It will help you use less resources during each cron run, because you will be able to choose how often will cron of certain modules be run.

In some cases this can result in slower email delivery. If you have a large number of emails to deliver (e.g. 100k newsletter subscription list), last user in the queue will receive the message significantly later than the first one. Also, all emails requested to be sent during that time, will be delayed as well.