Branding is a term that is often used incorrectly. Many designers and non-designers alike use the term "branding" to refer to "brand identity". I often do this too because branding is a more familiar term to most. If we want to be precise and get down to the nitty-gritty though, there is a difference between the two and they're not interchangeable terms.
Branding is the overall personality, presence, and feelings associated with your business. It's what and how people think, when they think of your business. It's portrayed in every touch-point of your business. Think about how the tone of voice on your website or social media, might give a certain impression of your company. Your customer service policy would also communicate something about your business.
The values, mission, and vision of your brand all contribute to your branding. Nailing your branding is important because it directs how you run every aspect of your business. It will also dictate your brand identity.
The brand identity is strictly the visual part of your brand, which includes your logo, color palette, font choices, patterns, and use of photography.
Branding is the overarching umbrella that encompasses the brand identity. The visual part of your brand is the quickest way that potential clients get a taste of your business. It can quickly repel or draw them in, which is why your brand identity is important.
The visual part of your brand is the quickest way that potential clients get a taste of your business.
Here's an example to put the two into context. Your branding might be based on the values of putting the customer first and creating a personal connection with them.
To communicate this, your brand identity might have the personal touch of a hand-drawn logo, paired with a warm, soft color color palette to create a welcoming and approachable feel. That's just one example, but hopefully it helps you understand the difference between the two, and see how a brand identity fits into the overall branding.