The distinction between being employed and self-employed affects your tax, benefits, employment rights and financial planning significantly.
Tax treatment
Employed: PAYE — tax and National Insurance are deducted automatically. Your employer handles it.
Self-employed: you pay Income Tax and Class 2/4 NI through Self Assessment. You must register with HMRC and file a tax return annually.
Employment rights
Employees have rights including: minimum wage, paid holiday, sick pay, maternity/paternity pay, unfair dismissal protection. The self-employed have none of these by default, though some rights apply to 'workers' (a hybrid category).
Benefits
Employees build up employment and benefits rights more easily. The self-employed pay NI too, but access to benefits like Statutory Sick Pay doesn't apply. Income protection insurance is often worthwhile for the self-employed.
IR35
IR35 is legislation targeting contractors working through a limited company who are effectively employed (e.g. working only for one client, under their direction). If caught by IR35, you pay employment levels of tax. HMRC takes this seriously.