Why would I buy one over the other -- what advantages are there for a jet engine? Is it a shallow water thing?

prop for lake fishing, jet for rivers or shallow backwaters.

you lose a lot of hp with a jet unit, 250 might be the same as a 175 propped (guessing).
jet is great for shallow water and rivers.

prop is a lot more efficient and faster.

most tournaments say no Jets as well so a prop boat that thinks like a jet is what most do

Have to be a bigger outboard for a jet than with prop

Jets are great for rivers with depths as shallow as 1ft. They can get you not areas that are generally unfishable by boats with props.

It’s all and only about running shallow.
If you want to get to places no prop rig can go then it’s worth a look.

Jets do not like deep rough water. Not at all.
Jets need a light hull to work correctly.

I run a jet on my scooter because where I go it can get pretty dicey. But I can’t carry five fatbodies and 400 lbs of beer.

But when it’s 2” deep to get to the back lake it’s a breeze.

When your jet is setup right, there's nothing below the bottom of the hull. If it floats, it goes. We jump oyster reefs and sand bars all the time to get from A to B, no harm no foul, just slide over and go. Can't do it in a fiberlass prop rig. Well, I guess you can but you won't like the results.