Will plants grow better with regular water or grey water?

Greywater will have a higher concentration of salt than fresh, or potable, water. Sources of salt usually include laundry detergent, soaps, residue from our bodies, and especially water softeners. Over time, if you add greywater to your plants, you'll increase the salt concentration in the soil, which is not good for most plants. It would help if you added fresh water or allowed rainwater to flush the soil. It would also help if you switched your detergent brand to one without any salt, and there are not many out there.

Other issues with greywater are pH levels, chemicals from any cleaning products you use like bleach or ammonia, and clogging solids like lint, hair, and scum. When distributed over enough area, most plants should be able to handle these issues. However, there are some plants cannot adapt to these conditions. So, yes, greywater will kill some plants.