Working in the industry, there seems to be confusion among Business Analysts as well as clients when identifying the differences between a need and a requirement.
A “requirement” is defined by the International Institute of Business Analysis (IIBA) as a condition or capability required by a stakeholder to solve a problem or achieve an objective, while a “need” is a high-level representation of the requirement needed. The need is the end result or purpose. It is the "why we are doing this".
“Business needs identify and define why a change to an organizational system or capabilities is required” - BABOK

Let’s take a look at these examples:
Needs (I need to...) | Requirements |
---|---|
build a big house for my large family |
|
get a BA job |
|
build my portfolio website |
|
Requirements are what need to be done in order to achieve the need or goal.
Another way to understand the difference:
Needs are high-level requirements that are segregated into lower-level and more detailed requirements.
Why it is important to understand the business need
Understanding the purpose of the project you are working on is very crucial to be able to elicit and analyze requirements. When you know why this system is being built, what the business need is for it, and what business impact it has, you will easily be able to ask the right questions when eliciting requirements and prioritizing them.
This will in turn help you reject requirements that don't satisfy the need. For example, we are building a system
for the call center department that allows us to capture customers' information. The need is to help
the marketing team reach out to non-paying customers and offer new products to convert them to paying customers.
Now that we understand why we are building this system, let's take a look at the following requirements and identify
what is necessary and what does not satisfy the need:
- The system shall save users' name, email and phone number ✔
- The system shall allow customers to post their photos ✖
- The system shall allow call center staff to send follow-up emails ✔
- The system shall require customers' emails in order to contact the support team ✔
Please note that the above example seeks to illustrate the importance of understanding business needs. While there may be benefits to including an function to allow customers to post their photos, this kind of feature does not obviously satisfy the need. Recognizing this difference will help you ask the right questions to the business owner, though - why do we need to allow our customers to post their photos? Are we allowing them to upload more than one photo? And so on.
If you have examples that illustrate the differences between business needs and requriements, we will all benefit from hearing your thoughts in the comments :)