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.
For example:
I need some company.
This suggests you have an emotional need, and "require" would sound far too robotic in this context.
Your examples:
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.
Best Answer
They are different. Firstly, let's see all three verbs in concern.
The verb requisition is used more for official purpose.
An example from OALD for this - The school was requisitioned as a military hospital.
The verb require has several meanings but in the sense you are guessing, it means request and expect (#2)
An example from WW for this - We require our secretary to be on time
The verb request is to politely express that you need something
An example from OxfordDictionaries for this - the chairman requested that the reports be considered
Now your examples:
Obviously, the latter sentence is more polite as it includes the word request.
Is there any difference in these both -- yes, there is.
The former one is just a statement that shows your requirement. On the other hand, the latter one shows you are politely asking for that requirement. To be a bit straightforward, I require a new car is ordering someone over I request for a new car is asking for a favor and doing so, that someone may oblige you by getting you a new car and not old! :)