Casual Conversation:
The volume of your voice is calm but enough for your voice to be audible.
Having a causal conversation usually focuses on many different topics.
Causal conversations don't need to be scripted or planned out.
Formal Public Speaking:
The volume of your voice needs to be loud enough for many people to cleary hear you.
Having to publicly speak means you should have a central message in your speech.
Public speaking are usually best to be planned ahead of time so you don't have to improvise.