Are you sure you want to get that deep into it
In short
Anemone: Attaches itself to the bottom of a surface by a adhesive foot. It is a single animal called a polyp. It is a solely soft bodied animal
Coral: Corals have or many not have a hard skeleton of aragonite, think like limestone. It is often not a single animal /say but is made up of many identical polyps but some corals can have a single poly. There are also hard corals and soft corals. Many soft coral have small rods in their soft structure to reinforce them. None of these rods or hard skeletons are seen in anemones.
http://www.ucmp.berkeley.edu/cnidaria/cnidariasy.html
Phylum Cnidaria
Class Anthozoa (anemones and corals)
..Subclass Alcyonaria
......Order Alcyonacea (soft corals)
.......Order Gorgonacea (gorgonians, sea fans, and sea feathers)
.......Order Helioporacea
.......Order Pennatulacea (sea panzies and sea pens)
.......Order Stolonifera
.......Order Telestacea
..Subclass Zoantharia
........Order Actiniaria (anemones and sea anemones)
........Order Antipatharia (black corals and thorny corals)
........Order Ceriantharia (tube-dwelling anemones)
........Order Corallimorpharia
........Order Ptychodactiaria
........Order Rugosa
.......Order Scleractinia (hard/stony corals)
.......Order Zoanthidea (zoanthids)