Regression through the Origin means that you purposely drop the intercept from the model. When X=0, Y must = 0.
The thing to be careful about in choosing any regression model is that it fit the data well. Pretty much the only time that a regression through the origin will fit better than a model with an intercept is if the point X=0, Y=0 is required by the data.
Yes, leaving out the intercept will increase your df by 1, since you’re not estimating one parameter. But unless your sample size is really, really small, it won’t matter. So it really has no advantages.