Branding
“The process involved in creating a unique name and image for a product in the consumers’ mind, mainly through advertising campaigns with a consistent theme. Branding aims to establish a significant and differentiated presence in the market that attracts and retains loyal customers.” Source
In this case theme being what I will refer to as style.
Logo
“Recognizable and distinctive graphic design, stylized name, unique symbol, or other device for identifying an organization. It is affixed, included, or printed on all advertising, buildings, communications, literature, products, stationery, and vehicles. Not to be confused with a brand, which identifies a product or family of products”. Source
Logo Design Basics
-Text Logo
Using typography in a unique way to create your identity.
-Symbol Logo
Using a symbol or image to create your company identity.
-Mixed Logo
Using a both a symbol and text to create an identity were they can work together or apart.
Theme
“Central, recurring idea or message that connects individual ads (comprising an advertising campaign) to produce a cumulative effect far greater than the total effect of the individual, disconnected ads.” Source
Brand Identity
“Characteristics or elements combined and expressed in a particular (often unique) and consistent manner.” Source