A common question, and the answer is less dramatic than certificate vendors suggest.
WHAT IS IDENTICAL
The encryption. A free certificate uses the same algorithms and key strengths as a paid one. Traffic is exactly as secure. The padlock. Visitors see the same indicator. Browser trust. Both are trusted by all current browsers.
WHAT DIFFERS
Validation level. Free certificates are domain-validated: they prove control of the domain. Paid certificates can be organisation-validated or extended-validation, which involve checks on the legal entity. Modern browsers no longer display this difference prominently. Warranty. Paid certificates often carry a financial warranty. In practice these are almost never claimed. Lifetime. Free certificates renew frequently and automatically. Paid ones are usually annual and renewed by hand, which means they can be forgotten. Wildcards and multi-domain. Available in both, with more options in paid products.
WHAT TO CHOOSE
For almost every business website, blog and shop, the free AutoSSL certificate is the correct choice. It is automatic, it does not expire unnoticed, and it costs nothing.
Consider a paid certificate if a partner, regulator or payment processor specifically requires a validation level the free option does not provide.