A logo is similar to a trademark. It's a visual symbol for the brand, and may appear anywhere the company wants to identify itself: On a website, product packaging, business cards and letterhead, advertising (digital, print, or TV), vehicles, promotional items, etc. A logo may be composed of type, custom lettering, or images (abstract like the Nike swoosh or pictorial like the Starbucks mermaid), or some combination of the above. A logo has strict specifications that must be followed regarding color, aspect ratio, orientation, etc.

A banner, as I understand it, is a header for a web page, or an advertisement embedded in a web page. A banner may include a logo, but it may also contain other words or images, and unlike a logo, its design may vary depending on its specific purpose.

As I recall, they are usually termed 'banner ads'. I have read of 'banner' being a full-width headline in a newspaper, which may be where the 'banner' in 'banner ad' comes from.

As for 'a company's banner', I think that is more akin to a medieval king's banner (flag or standard with his heraldic device on), where the banner is used as a symbol for the company. A part makes whole, or something like that.

Why is someone who works for a web design company asking about these basic terms? My first thought was that it was a spammer.

But since we're answering the question, in addition to the definitions already given "banner" can also refer to a decorative device highlighting a small amount of text. It can hold a logo, but usually not.

In my experience (mostly printed materials) this is the usual definition of "banner", but maybe it's different in web design.