If you want to get technical, Watts are a unit of power and mean 1 joule (unit of energy) is dissipated every second. So a 350W can supply up to 350 joules to the machine every second. A 500W can give 500 joules every second.
Quite literally, it is more powerful.
A more powerful unit will have more connectors, since it will be expected to handle greater loads. No point having a 1000W Supply with only the ATX cables, you would never be able to use its potential and would be useless to anyone who does need that much power.
You need a more powerful PSU, otherwise the machine will stay sluggish and over time the PSU being under heavy load can damage the components.