A tarp isn’t more or less rain proof than a tent. It’s definitely how you pitch it
You might find yourself with more room in a tarp than in a tent, and being able to pitch it so you can see out means you might feel less claustrophobic
I don’t think there is a clear winner in all situations, most folks with both options and the ability to pitch a tarp modestly well (or better) agree that both have their strengths so you pick the best one for situation you’re likely to encounter
A tent is reliably the same, but it means it’s not flexible. A tarp and bivy give you more flexibility, but to get it reliable the same takes more practice and skill, and can be slower to pitch too.
e.g. I slept under a porch and in a cave in March in the lakes, both in a bivy. Tent wouldn’t haven’t been possible in a tent (unless it’s freestanding)