Pickles are usually vegetables preserved in vinegar and salt, sometimes with sugar and spices. They tend to be sharper and more direct. Relishes are often chopped finer and can be easier to spread. Chutneys are usually thicker, often cooked down with fruit or vegetables, sugar and vinegar for a richer, sticky finish.
For sandwiches, pickles bring crunch and tang, relishes spread neatly and chutneys add depth and sweetness. If you are making a ham and cheese sandwich, a mustardy relish like piccalilli adds bite. If you are building a cheese board, a sticky onion chutney style condiment can sit alongside crackers and blue cheese.
The easiest rule is this. If you want crunch, reach for pickles. If you want spreadable tang, choose relish. If you want rich sweetness with sharp edges, go for chutney.