What is the difference between a taught and research MSc and a research MPhil degree? I'm in my senior year of Bsc Civil Engineering, in which I've started applying towards my grad studies in the  U.K.

In searching for the courses I've seen an MSc of Engineering Science by Research (2 years long) and MPhil of Engineering (1 year long). The first is a kind of taught and research degree while the second is only a research degree.

What are the differences between these kinds of degrees, academically? Which of them is a more prestigious degree?
Many take an Mres with no plans to go onwards to a Phd whilst an MPhil tends to be the precursor to a Phd programme (both tend to have some taught component, but sometimes both can be just pure research, there is no one size fits all as each programme is unique).  

When joining a Phd programme you will always be put on an MPhil first whilst the same is not true for an Mres.  That is not to say though that you cannot extend your Mres project to a Phd if suitable funding and supervision is available.

As an engineer the only reason to do either of these research programmes is if you actually wish to do research.  If you just want an advanced accredited degree its better you look at Msc or MEng programmes.

Neither degree is more "prestigious".  They are both designed to teach you how to become a researcher.  If you want to do research in engineering though you will need to do a Phd or spend your career with an Mres working under the supervision of Phds.