Annuals live for one gardening season and then die. So, in the fall these need to be removed. One exception is sunflowers that can be left for birds to enjoy. After pulling up the plants that have died, compost them. Do not add any diseased plant parts to the compost pile.
Perennials are plants that grow back year after year, assuming they have proper conditions and care.
Fall is the time to cut back dead growth on herbaceous perennial plants. Dead leaves and stems on the ground can invite pests to overwinter, so cleaning them up now is important. You do not need to cut them back to the ground since leaving a couple of inches of growth is helpful for trapping snow for insulation. Unless a perennial plant is tender or not well suited to survive in your hardiness zone, you should not need to mulch. Any mulch placed over the beds during the spring or summer, along with upcoming snowfall, should protect the plants sufficiently.