There is a difference, and they are not always interchangeable.
In the most basic context, either are ok:
- I require a screwdriver.
- I need a screwdriver.
The difference is that "require" is far more "matter of fact" than "need", which is used where there is a more human, emotional context.
I need some company.
This suggests you have an emotional need, and "require" would sound far too robotic in this context.
If you want to be a successful teacher you
will need patience.
Either would probably work here, but I prefer "need". A "requirement" tends to refer to something that needs fulfilling once, otherwise one normally specifies that it is a regular requirement. A "need" seems more suited for something constant, or ongoing. I found the word "will" unnecessary in this context as it seems to be an immediate need.
I'm thirsty. I need a glass of water.
Again, either could work - "require" just sounds too robotic. Even though food and drink are daily "requirements", the fact that you talk about being "thirsty" which is a feeling makes "need" seem more suitable.
I need to speak to you before his arrival.
"Require" doesn't interchange in this context. We don't say "require to..." - we would say "I have a requirement to..." In a formal setting, you could perhaps say "I require a word with you before his arrival".
They require the information to be sent as soon as possible.
Either would work - and perhaps "need" would be most people's choice as it is common, informal speech. As an example of where "require" does work though, this is a good one because somebody else's requirement is being passed on second hand, which is very distant and emotionless.