A brand is a promise, a logo is a reminder of that promise.
From your organisation's side, your brand is made up of three things: what you promise, what you deliver, and how you package the two together. Excellent brands make a distinctive promise which they fulfil consistently by delivering exactly what they said they would deliver, and present this with a combination of written, spoken and visual elements which communicate the promise clearly, credibly and relevantly.
From the public's side, your brand is the sum total of all the things people remember about it.
The logo is a visual device which identifies your brand to help the public to remember it. If your brand has premises, vehicles, printed materials or its own website, a logo is an important tool. If your brand works exclusively through word-of-mouth, or is an aural brand, such as a radio show or podcast, the logo will be comparatively less important.