This one is by request, but it does intrigue me, and no, I will not be attempting to define "irregardless".
In fact, this comparison is really regard v. respect, since both words contain modifiers for the negative ("ir" and "less"). Regard has a complicated etymology. Regard originally derives from Indo-Europen "wer", through Middle English "warde" (and its variant spellings) through German "warten" to Old French "garder" for "to watch". Regard, then, literally means "to watch again", however, figuratively, it means "to pay attention to" as it derives from to the idea of watching again and again being that you don't stop watching, or paying attention. The word evolved into "to think highly of and/or with a particular feeling" as if when you are truly paying attention it is because you would think highly of the person or have a particular feeling that would inspire the attention. Therefore, regardless naturally means inattentive or unmindful.
In contrast, respect comes from the Latin "specere" for "to look", and thus, respect means "to look at or consider again" and by extension, finally, to mean "deference". Therefore, irrespective means without a second look or thought.
So, the real issue is given the closeness of the etymologies, how does respect differ from regard? Regard has its origin from watching with a purpose, hence the evolution through guarding and paying attention and imbuing the watching with a higher sense of worth, while respect is just looking with no purpose, and then as a result of what you see, paying closer attention and giving the look higher worth. The intent of the initial observation is different, leading to a different purpose for observing, although quite probably leading to the same type of ultimate observation. However, there is a separate distinction through the implied usage. Regard looks at the physical characteristics, again, the reason for guarding or watching. Respect looks at a quality of a person, the reason for giving a second look. Now, a quality of a person may be a physical characteristic (a pretty smile, a scar), but it was not the reason for having to watch the person. Therefore, we give regard to pedestrians at a crosswalk, and we give respect to the police car stopped on the side of the road. The pedestrians are a group of people who need protection against traffic and therefore give continual watching to, while we have no particular interest in the police car except for what the officer may subsequently decide to do which could affect us, therefore, we pay more attention until its relevance is moot. We regard beauty as an asset, and at some point we hope that others will respect the person for more than pretty looks. To take some common and easy ones, we are told to respect our elders, implying that we would not give such notice on first glance, but we should look deeper to find something worthwhile and therefore, worthy of deference. We don't regard our elders. That just sounds odd (like a malapropism), unless they are feebleminded and need elder care. Then, it's appropriate.
When we return to regardless v. irrespective, however, these words are generally used on a meta-level to regard v. respect, as in the fact or quality of what should be regarded or respected. Regardless of the fact that it was physics exam, the student answered the essay questions with dissertations on economic philosophy. True story. Irrespective of his desire to maintain his 4.0 GPA in economics, he submitted the essay for the physic professor to grade. Regardless has the idea of ignoring something to which you should have paid attention, while irrespective is dismissing something to which you had no need to pay attention. Regardless of the weather, my friend and I go walking every morning (almost true, but not due to the weather). Irrespective of his 7 y.o. daughter's whining, he goes to work every day. Ok, that might be a little harsh. Depending on the parent, it could just as easily have read, regardless of his 7 y.o. daughter's whining, he goes to work every day. So, depending on the person and the societal norms of what we should "regard", and even just cautious politeness, regardless has a broader usage. Irrespective, then, is almost flippant, as well, the lawyer in me prefers "notwithstanding" as a more generic, and perhaps obscure substitute. Irrespective of the judge's counseling, Plaintiff's counsel proceeded to attack the witness's credibility on his extra-marital affair. Only Plaintiff's counsel would actually presume to defy a judge so blatantly. For the rest, it would be regardless of the judge's counseling, the attorney continued to zealously represent her client by cross-examining the witness on his extra-marital affair to attack his loss of consortium damages.
So, irrespective (or perhaps regardless, as you see fit) of what you think of my analysis, perhaps we should work to err more on the side of regardless.
Ed note: I started this comparison over a month ago, but it took some time to really process the subtle differences, and I couldn't extract myself to work on any other words until I finished it, hence the extreme delay. Again, thank you for your patience. Hopefully, other words will not create such obstacles...