OK, let's try to figure out where the installation failed. You'll need to use phpMyAdmin to check the following database tables.

1. configuration_group. Should have a record with a configuration_group_title of VAT4EU Plugin.
2. configuration. There should be 9 records with a configuration_key LIKE VAT4EU%.
3. admin_pages. Should have a record with a page_key of configVat4Eu.
4. address_book. Should have two new fields named entry_vat_number and entry_vat_validated.
5. orders. Should have two new fields named billing_vat_number and billing_vat_validated.

If you could perform the above database checks to see where the installation stopped, I can suggest what step to take next.