They can look the same, being more or less cousins, but they aren't. Sabres are slashing weapons, rapiers are piercing weapons. Sabres are fairly weighty to enhance the power of the slash while rapiers are intentionally light to allow quick control of the tip. Sabres do have sharp points just as rapiers have sharpened blades, but it is not the primary function and it's obvious if you handle the two weapons side by side.
I have a heavy rapier that from the side is sabre-sized. I could use it to slash, but it would do badly because it is too light. The shop had an almost identical sabre which was half-again as thick and far more unweildy (at least for the tactics I learned from combat fencing with epees) but it seemed like it screamed at me to swing it at arms length to smash something.
Never understimate the power of leverage. That slight thickening of the blade doesn't mean much at the hilt, but 20" from your hand it starts seriously cranking up the momentum & inertia.