Tax number: for companies doing business exclusively in Germany, this is sufficient.
A tax number consists of 10-11 digits, usually separated by two slashes. The tax number is issued by the tax office. Already by moving the tax number can be changed by the respective local tax office. (Example: 00/001/00001).
VAT ID: for companies that conduct cross-border (EU) business transactions, this is mandatory.
A VAT ID identifies a company within the EU. The VAT ID number must be applied for at the Federal Central Tax Office (BZSt), stating the tax number. A VAT ID is only available for companies or self-employed entrepreneurs. For business within the EU, a VAT ID must be available. It begins with the EU country code, then a series of up to 12 digits (e.g. DE123456789).
The VAT ID must be stated in the imprint on the company website (§5 TMG), otherwise, this may result in a warning.
If a VAT ID is provided, the time-consuming double taxation procedure is not applicable, i.e. if both companies have a VAT ID, the supplier of the goods or services may not charge VAT and therefore does not have to pay it. Only the buyer or customer must pay the VAT in his country to the tax office there in accordance with the regulations applicable.
The correct VAT IDs of both companies must appear on the invoice!
Tax ID: personal for a natural person, not relevant for companies.
The tax ID is personal and is issued by the Federal Central Tax Office. It consists of 11 digits, it is requested by the tax office, but also by banks, family funds, pension funds, or also the Federal Employment Agency, e.g. 99 999 999.