Using the web interface or using Drush cause different result? What is the difference executing an upgrade from Drupal 6 or 7 to Drupal 8 using the web interface or it's better to use Drush?

If the database is heavy, using either of them would have different result?
From the technical perspective, the PHP configuration can be different for web server (Apache) and when running from CLI (drush), since different php.ini is loaded, therefore different limits are applied, sometimes your PHP could be different.

If you're non-technical person and it's one time (simple) process, it's better to use web interface, which is recommended way to upgrade your Drupal. The size of database doesn't make any difference, since the batch operation can perform long operations, but it may fail in the middle if you lost your connection.

On the other hand if you're repeating the process many times in order to test it properly, Drush may be a better solution, since you can test the process in more efficient and consistent way especially across multiple environments, this is especially useful if you've continuous integration in place (such as deployments).

So it's really depending on your needs.