I know that, in general, there is a difference in strictness between the two categories of protective measures, but am having difficulty in explaining the difference between the two.
Gezeira means the rabbis prohibited specific acts that could likely lead to Torah prohibitions -- e.g. if you're reading by the light of a low-quality candle, you may forget yourself and play with it to get better light.
Shvus are categories of activity -- such as "business transactions" and "healing" that the rabbis banned. There had to be a technical clothesline on which they could be hung, e.g. "we banned all business transactions as they may lead you to write", "we banned all medicine [unless for the seriously ill] as it may lead you to grind herbs", but it appears that the broader spirit of Shabbos came into play to a significant degree. It wouldn't be a day of holy rest if you were overwhelmed with business transactions (even if you didn't technically write anything) or obsessing with every proposed cure in the book over your slightly sore pinky toe. So they banned the whole category. (Even forms of medicine that have nothing to do with herbs -- let's say something like traction. They banned the whole category called "medicine.") Halachically, a shvus is generally stronger than a gezeira, e.g. there are circumstances that will override a gezeira but not a shvus. This makes sense when you think about it -- if I remember not to play with the candle, a shabbos where I read is still shabbos. But a shabbos where I'm negotiating stocks is not "remembering the day to keep it holy."