Forward will do just that, forward emails sent to that alias e.g firstname.lastname@example.org to a set user. You would give this to a user if you only needed to recieve emails not send as them.
Giving someone full access gives that set user full access to the mailbox so they can then add it to their outlook and use it as if it was their own. You'd do this if you wanted full access over the mail box, Send / Recieve
Edit: another one to chuck in the mix is send as access which allows a user to send from a alias but not recieve mail from them - This speaks for itself, you'd give this to a user if you wanted them to send from that alias e.g email@example.com when sending out bulk emails