Dear Kenty
1. Check for .htaccess files in the root folder of your site. If any make a backup and delete from server.
2. It might be possible that wp-config file had different table prefix then what is provided by current hosting. Compare both old and new wp-config files and make sure only 4 values are different viz. dbhost, dbname, dbuser and dbpass.
3. Navigate to your site by FTP to wp-content and rename plugins folder. Create a new folder named plugins which shall be blank. (Just in case plugins are creating problems)
4. There might be error with database. Usually wp recognises the installation to be new if relevant tables are not found in database.
Recommendation: go for fresh install of Wordpress on your server (back up all data and databases). Upload theme files via FTP for matching earlier appearance. Use Wordpress exporter tool to move content between servers.
Let me know if you are still troubled :)