Is there a difference between wind/water proof shoe covers you'd use in bad weather and the ones you would use for time trial?

And if they are different, what are those differences specifically?

yes.

Time trial shoe covers are just thin lycra for aerodynamics. Winter booties are thick neoprene like scuba clothing. There are some inbetween booties for less cold weather too.

this thread has potential.

i think the word you're looking for is "wetsuits" lol. some winter booties are like wetsuit material (generally thinner) and NOT waterproof. the thicker & usually more expensive waterproof booties tend to have an extra layer on the outside similar to a rain jacket that makes them waterproof/resistant.

doubtful

Yeah, obviously winter ones are going to be thicker/more insulated. It was more the thinner ones for wind and rain that I thought there might be similarities.

Booties come in all shapes, sizes and thickness, buy they ones for your situation.