There are pros and cons to both.
Embroidered items can cost a little more, but their perceived value is much higher. The imprint will also last a lot longer since the embroidery won’t fade away with the passing of years and tons of washes.
You should also consider the fact that an embroidered logo literally stands out from your item. Screen printing sets a very fine layer of ink to your product; embroidery has more mass to it so it will rest slightly higher than your item.
Also the majority of embroidered items allow for you to have multiple colors without having to pay extra. The allowance of extra colors makes embroidery ideal for schools and organizations who have a multi-color logo and don’t want to simplify their color palate