Embroidery is basically defined as decorating fabric using a needle and thread. Screen printing is print making in which a design is printed on a screen of silk or other fine mesh with blank areas coated with a sealed substance and ink is forced through the mesh onto the printing surface usually a t-shirt.
Advantages of embroidery:
All fabrics can embroidered, where only smooth fabrics can be screen printed.
Embroidering small quantities (<30) is usually more cost effective than screen printing.
Embroidery often outlasts the garment and will not fade or wash away.
Large letters (names, etc) can be more cost effective than screen printing.
Embroidery generally looks more professional than screen printing.