There are two main things I can think of, the first is the autofocus. On a DSLR, the mirror reflects light on to a Phase Detect Auto Focus sensor while you are looking in the view finder. While PDAF isn't as accurate as contrast based detection (which can be done with a standard CMOS sensor) it is much faster. Since mirrorless lack the mirror, they generally use CDAF which is slower. This is probably the majority of the difference.

The other thought would be shutter. If there isn't a physical shutter on the particular mirrorless, it is going to have to use an electronic shutter which may take more time to clear the CMOS sensor before exposure, but I would expect this to be a significant contribution.

Size could also be a factor as mirrorless tend to be smaller, and thus have less room for electronics. It's hard to make electronics small, fast and power efficient, so they may have elected for slower electronics to keep small and battery efficient. Older cameras had speed, but they may be doing additional image processing now and higher resolutions now would require faster processing as well.

Update: The information you supplied about the burst makes me think it has to either be CPU or memory card access speed related. If the burst is fast, but between shots is slow, then it means that it is the actual image processing and storage that is the bottle neck. Generally, burst shoots in to a buffer and then images are processed out of that buffer and then written to the memory card. If burst is fast, then it is making it in to the cache quickly, which rules out focus or shutter issues on that particular model. The only thing left is the process of clearing the buffer out, and that's image processing and storage.

It's fairly typical for the camera to wait for image processing and storage to be completed before moving to the next image when not firing in burst, so this would seem to support that assumption as well. If comparable cards are being used in both, it would be a question of the speed of the actual card interface in the camera and if that is comparable, then CPU is really all that is left, at least of the things I can think of.