It is necessary (=must be done) to peel vegetables because the outer skin is not edible/for hygienic purposes/because they cook more quickly without the skin.
It is needed to peel vegetables.
A person does not scratch off the skin from a potato with his nails, nor rub it with sandpaper. There is a special kitchen tool called a vegetable peeler. I bought one because it is needed (=required for the purpose of peeling) when you peel vegetables.
"It is necessary for me to go to Tokyo tomorrow to discuss a business contract in person."
"It is needed to go to Tokyo tomorrow."
"His Japanese is rusty. Where is the office copy of the English-Japanese dictionary? It is needed for when he goes to Tokyo tomorrow so he can take it with him."
With regard to the 'urgency' your friend mentioned:
I might say, "Can someone please tell me where they've put the umbrella!? It's needed. Jenny's got to go home and it's pouring down outside."
Here, the implication is 'it's needed right here and now', compare, "A rain coat is necessary for when it rains." (So I'll buy one next time I'm at the shop - no implication of urgency. This is merely a piece of advice.)
The above are some of the ways in which they are different.
compare:
I need to have a high level of proficiency in Japanese before they will consider me for a job in the Tokyo office.
A high level of proficiency in Japanese is needed before they will consider anyone for a job in the Tokyo office.
It is necessary to have a high level of proficiency in Japanese before they will consider anyone for a job in the Tokyo office.
Here, 'is needed' and 'is necessary to have' both have the meaning 'is required'. A slight difference in meaning is:
'is necessary to have' : some external authority demands this as a prerequisite, otherwise, the person is not eligible.
It is necessary to have good grades in science courses to enrol in the Medical School.
'is needed' : is required in order to perform something adequately/proficiently. There is no suggestion that there is a 'rule' saying you must have it.
A good suntan lotion is needed in the tropics to prevent burning and skin cancer.
However, by rephrasing a sentence, either could still be used, as I indicated at the start of this post:
Good grades in science courses are needed before they will admit you into... : here, 'before they will admit...' serves the purpose of stipulating that it is a rule.
Similarly,
It is necessary to have a good suntan lotion with a high SF factor if you want to prevent sunburn.
