Volts and Amps and Ohms, oh my!
They are related. Quick and simplified: Amps is the amount of electricty flowing. Ohms is how the wire blocks the electricty flowing. Volts is how much pressure is behind the electricity. Watts is the amount of work done by the flowing electricity. Ohm's Law (and Watt's Law) describe the relationship between volts, ohms, amps, and watts.
The OP question about what the difference is between various '8 watts'. I have a 100 watt electric blanket. It keeps me very warm, and doesn't burn me. I have a 100 watt soldering iron. It can burn holes in things, and melt metal (solder). Difference is how much heat in how little space. Larger coil is usually higher ohms, and will heat up more ejuice at a time, but not too hot, and not too fast (bigger pot simmering). Smaller coil/lower ohms will heat up less ejuice at a time, but hotter and faster (smaller pot boiling). Some people like it one way, others like it the other way.
Battery capacity: some batteries are rated in watt-hours, but most batteries are rated in milli-amp-hours (multiply by voltage to get the milli-watt-hours). One of my batteries is rated 8 watt-hours, implying that I can run that 8-watt coil for 1 hour before my battery needs charging. 3.8 volts / 1.8 ohms = 2.111 amps, so I'm guessing my battery is 2100 mAh, meaning it can supply 2.1 amps at 3.8 volts for 1 hour.
A rough guide to battery life: 100 mAh gives the average vaper 1 hour of vaping time. I'm a slightly heavier vaper, and start a fresh battery in the morning, and it needs a charge by the time I go to bed.
Real world experience: In theory, practice and theory are the same. In practice, they're not.