Olga, it's an interesting question. There's no real difference. These words are useful discourse markers or signal words that shift attention to something that was either stated earlier or is related to what was stated earlier. Although in speech 'regarding' is a more fluid transition from one topic to another - that 'ing' makes a difference. For example: you and your manager are having a casual conversation. S/he is talking about work. You want to draw the attention to the your recent increased work load, and the possibility of getting paid for it. 'Regarding the extra hours I've been working recently.....' 'With regard to' stops the flow of speech or text- which you may want. With everything it's a matter of context. I recommend a language corpus or collocation websites. With Russian I use the Corpus and of course Yandex. Good luck!

I'm not saying that Logan is wrong, he is mostly right, but with respect to certain academic subjects he is. In academic literature you'll often see use in the middle of a sentence, the way I did in the previous sentence. It's also common in mathematics and math heavy fields, especially when describing calculations such as derivatives, to see it in the middle of a sentence.

This is a phrase you'll usually see lining the beginning of a sentence. <____________> the project, we will require all documents submitted by the end of the day. You can place any of those in there except for "as regards." I don't think I've ever seen that used in a sentence like that. These phrases are just ways to address a topic at the beginning(most likely) of a sentence and elaborate on it. It's about using it for clarification.