When is it better to use the word 'Irregardless' rather than 'regardless'?

Never, as the Merriam-Webster dictionary appropriately points out: https://www.merriam-webster.com/dictionary/irregardless

Although irregardless has become fairly common, many sites point out that it is a double negative since the prefix 'ir' and the suffix 'less' mean the same. Using the word changes its meaning from 'without regard' to 'with regard' under strict grammatical rules. In common language it might pass unnoticed but using it in an essay is likely to get your work downgraded.

As my daughter would confirm, I'm pedantic, so would avoid using the word myself.

Grammarly has an interesting article here which debates both sides of the argument: https://www.grammarly.com/blog/vocabulary/irregardless/

Anyways, irregardless of what Grammarly says, I will unmercilessly continue to orientate my language towards speaking English proper.

It's a bit like the flammable/inflammable situation, but I am nevertheless firmly on the side of the pedants.