Public liability insurance covers you if you damage a client's property or someone injures themselves on your premises or because of your business activities. If you work from home but visit clients for part of your job, you'll want public liability in case of any accidents caused by you. For example, if you spill your coffee on your client's laptop or they trip over your laptop lead.
Product liability insurance covers you if you supply a product that makes a customer ill or causes injury. If you sell to the public, it's important to have product liability cover, as your product could cause harm to the customer months or even years after the product was purchased.