One of the key distinctions is about control. You pay a plumber to fix your tap, you do not give him instructions about how to do it and if he decides to send an assistant plumber to stop the drip he can.Does hiring a plumber/electrician/gardener count as employing them? I'd have thought they are the same as any other self employed contractor.
If the carer could not send another carer in her place she is more likely to be an employee. That means the employer has to have insurance, a pension scheme and so on and can direct how things are done.
Of course sometimes the line is blurred but that's an outline of the basics.