Emm it's a bit difficult to explain so I'll give some examples to make it easier for you :) For example: can I ask you for the restroom? (This would be a formal question in a restaurant maybe) And request it can be like this: Sorry miss, but we can't afford your request because you owe money to the bank. Emm the verb ask it may change the formality level in the way it's used... I hope I helped you :D
If you have to send a paper form to the government to get something done, that would be a request (form).
In a less formal setting you would normally ask for something instead of requesting it.
Unlike 'ask for', you can use request also as a noun. If you ask someone for something, that is a request.
Using request when 'ask for' would have been more appropriate is probably better than the other way around.
Police officer: Sir, please move your car as it is blocking traffic. Person: Is this a request or am I obliged to do so? Police: Neighbours have complained about your car, which is why I am asking (you this).