"Spring Water" should be chemical-free because it comes from natural, underground springs, and is not processed in any way. It may, however, contain various minerals that are present naturally in the location of the spring. Real, natural mineral water (that you buy in a bottle)comes from natural springs in mineral rich areas.
"Tap Water" has been processed at a water treatment facility before it ever gets to your tap, and has chemicals added to make it 'safe' for drinking (meaning without too much bacteria in it). Numerous chemicals are usually added, at least in the USA, such as chlorine, and often flouride as well, which is supposed to be intended to prevent tooth decay.
Lots of bottled water is tap water, sometimes filtered to remove some chemicals and impurities, sometimes not. Read the label, it should give more details.
Also, if you happen to live near a natural spring, you can take empty bottles there and collect your own spring water for free. I live near a spring and do that quite often. Make sure your bottles are CLEAN first! Very, very clean.
Also, not ALL plastic bottles are dangerous, only the ones containing BPA (which is more likely to be found in cheap, disposible plastic bottles). You can buy sturdy plastic bottles with no BPA in them, just read the label and make sure it says "BPA Free".