Usually the main difference between a spin bike and a stationary bike is the amount of technology involved. Spin bikes normally have fewer electronics, meaning that you are in control of your own speed and resistance the whole time. However, spin bikes often have more parts of the bike that you can adjust to ensure that your body is in a good position.
If you want to simply be able to push a few buttons and have the bike's resistance change automatically (hill or interval program), then a stationary bike might be a better option. Many stationary bikes also have the ability to measure your heart rate, which is a really good bonus as well.
It sounds like you could probably try the spin bike again and maybe try to adjust the handles and the seat position to make it better for your back. But, if that doesn't take the pressure off of your back, then go for the stationary bike. Other factors can come into play that could cause pain in your low back, such as too much or too little resistance, poor posture, or an unstable core.
You really just have to weigh the pros and cons to see what machine would be best for you... I personally like the spin bike better, but it's completely up to you