Learn English – “Sometime” vs “some of the time”

I'm both a native speaker and a published author, and if there's a difference here at all, I'm not convinced that I know what it is. It would be very subtle indeed, and they are basically synonymous.

All that said, you are correct that, in spoken English, for a longer time is more frequently used. I would typically avoid period of time altogether unless (1) the period is something that I wanted to emphasize and (2) the amount of time is not known. Even then, I would normally only use it in rather formal statements:

There was a period of time when Danish politicians would have jumped at the chance for this kind of press coverage.

We will not engage in bilateral relations for a period of time to allow everything to cool down.

Will my driving demerits expire after some period of time if I don't get any more tickets?"

However, in spite of my hesitation to use it normally, I want to emphasize that people do sometimes use it in informal speech, and that it is synonymous in every context you would expect it to be, so I wouldn't worry too much about this particular distinction.

"In style" primarily means that something is currently fashionable. For example, you might say that a particular item of clothing is "in style".

"With style" typically means that some thought, design, or flair has been put into something. For example, if you do something 'with style' it may be impressive, or look good. Note also that the expression "with style" is sometimes used humorously if something is unexpectedly a spectacle - for example, if someone fell over in an exaggerated or embarrassing way, perhaps knocking things over, someone might say they fell over "with style".