A hand mixer, I’m going to assume you mean an electric hand mixer, also called an electric beater, is used to make batter, merengues, and to whip cream in whipped cream. The ingredients are placed in a bowl, and then you use the mixer to “mix” the ingredients. You can often do it by hand also with a spoon or beater depending on the ingredients, but it takes a lot of strength to do it manually to get the same results.
A traditional blender you put the ingredients in a container (usually made of thick glass, but not always) and then you you completely blend all ingredients together. Blenders can shred or purée solid ingredients like fruits or vegetables, while the hand mixer cannot do that.
There is a such thing as a hand blender, also called an immersion blender. This will also break up ingredients that are solid, but they need to be soft already. Like stewed tomatoes you could hand blend into a smooth sauce, or pinto beans into refríed beans, etc. Like the hand mixer, the hand blender you have the ingredients in a bowl or pot, and you place the appliance into the ingredients.