When should I apply a multiple comparison adjustment?
The p-value adjustments corrects the inflated number of significant results when performing many significance tests.
If you have a test with 5 % false positive rate and but you run it several times the chances of getting a false positive results in any of these tests is far greater than 5 %. For example repeating the test with 5 % false positive rate 20 times means that there is a 60 % chance of at least one false positive result.  

You can see this in the tutorial you have linked to in your question, where the number of significant differences goes down from 26 without correction to 18 with correction.  

Therefore, if you are performing multiple comparisons you should do a correction. The problem of inflated p-values will occur when you are performing many tests even if the tested variables are independent.