A biological washing powder contains enzymes which break down stains and dirt whilst in the wash. A non-biological washing powder does not contain these enzymes and this is the major difference.
Sometimes the enzymes may be artificially created so that they are more able to remove stains, particularly ones which are very deep and dried on to the fabric.
Some people with very sensitive skin may find it better to use a non-biological powder, although this may not be as effective at getting rid of stains !
In terms of the environmental impact, neither system is perfect. Whilst the enzymes contained in a biological wash powder may take longer to disperse within the environment, the temperatures for washing can be lower than for a non-biological powder. The non-bio powder could take a higher temperature to do the same wash, which uses more electricity and water and this also has an environmental impact.
You can use either in a washing machine.