The difference is negligible. Running water over toothpaste only increases the likelihood of toothpaste rinsing off the brush. I run water before applying. It "refreshes" the brush. Like washing an apple before consumption.
I'd say it's more like wetting a paintbrush vs using it dry, it refreshes the bristles. And I'm surprised so many people in here are saying they wet the toothpaste, as you said the only potentiality there is you losing some of that toothpaste, so why bother.