So 650B & 650C have 584mm diameter & 571mm diameter respectively.
As a general rule, bigger is better
For road riding it's all about rolling resistance and bigger wheels will gloss over bigger bumps. Bigger is smoother therefore faster & more comfortable.
As for your height, for road the only thing you really need to worry about is toe overlap. If you go to a bike shop & try out a bike your size with normal road wheels and have no toe overlap with the front wheel, you'll be fine with anything smaller as they are 700c which is 622mm diameter. If you do have overlap then the amount of overlap will tell you if the decrease of 6.5mm in radius is worth it (b vs c); note that 700c vs 650b already has decrease of 19mm in radius.
The final thing is, 650b is just more popular! That just means you'll have an easier time finding tyres & other spares. 650c is a niche as there are no real benefits except for sizing.
When I'm not out riding, I'm in dreaming in Solidworks.
The designation is kind of a road and mtb separation.
The B wheels are measured, or designed in manner for american standard (inches).
The C wheels are measured or designed in a manner for metric standard (mm).
Thus, the C are narrower, slightly larger circled, and allow you to run thinner tires.
In more simple terms, 650c is road specific, not suitable for mountain bikes, and is a standard here in the USA that is no longer used. 650b is more common in Europe than the USA. Here it is referred to as the French standard. It gained some traction in this market, but has been falling out of favor once again. Plenty of tire choices in 650b for road and mountain.