An extender listens to your router and repeats what it hears. That is simple and cheap, but the repeat happens on the same airspace, so the bandwidth available to devices behind the extender is roughly halved. It fills a dead spot without making the network faster.
A mesh system replaces the coverage problem rather than patching it. Several units cooperate as one network, often with a dedicated radio link between them, and they hand devices between units as you move. To your phone it looks like a single, large network.
The practical difference shows up when you walk around. With an extender, devices often keep talking to whichever unit they joined first, so the signal degrades as you move. Mesh units actively steer devices to the nearest node, which is why video calls survive the walk from kitchen to office.
Neither fixes a slow internet line. If the connection into the building is the bottleneck, better coverage will spread the same limited speed across more rooms, and speeds will not improve.