Apparently, there’s a popular belief among English teachers in China that “continue to do” and “continue doing” mean two different things. According to them, “continue to do” means “start doing something again that was previously interrupted”, whereas “continue doing” means “to carry on the same thing you have already been doing”.
Native speakers make no such conscious distinction. The one or the other form may sound better in a certain situation or be more preferred in a certain part of the English speaking world, but this has more to do with the rhythm of the language rather than there being a strict dictionary definition separating the two. For example:
I will continue improving my skills.
mean completely the same. Some speakers may prefer the first variant, some the other one, but none of these will tell you whether you stopped improving your skills and started again or not before.