The carbon dioxide causes the cookie to rise. After baking, it falls. Omit the soda and you can expect a puffier cookie that takes longer to cook. It'll be good, but it'll be a different cookie. Fat is probably the most critical ingredient for turning out a great cookie.
Good bakers prefer butter because it tastes best.
Butter can be tricky to use, however. It melts in your mouth. The butter will be melting onto the cookie sheet before the starch and protein in the cookie have time to set. Shortening and margarine have higher melting points and are made up of different fatty acids, so they take longer to melt. Cooks have a wider margin of error as a result, but the cookies don't taste as good.