If one look at the kanji he should say there's no big difference between them.
If a difference should be pointed out is in the kaji 切.
大事 daiji is "a big fact", i.e. a fact of great importance.
大切 taisetsu uses the kanji of kiru to emphasize 大. Much like it happens in 親切 shinsetsu, 痛切 tsuusetsu, 哀切 aisetsu, 適切 tekisetsu, 懇切 konsetsu.
So, if we look at the kanji, we have "overtly big" or "extraordinarily big".
Setting aside the emphasis there shouldn't be a real difference, however the use of taisetsu is narrower.
It's surely true that you could say "jaa, sore wo taisetsu ni ne", while you would say "o-daiji ni", but this doesn't mean you'll use daiji in formal context only, and one can safely say "soredewa o-karada wo otaisetsu ni", much like he does using daiji.
Then the difference isn't in the formalilty or in the politeness (but I'll talk later about this).
If you try to google "大事なポイント" or "大切なポイント" you'll get the same number of results, more or less, and you'll see similar contexts.
When the context allows it, one can translate daiji as "important" or "relevant" and taisetsu as "precious".
On the other hand, if precious is "too much" for the context, then we can see/feel that taisetsu is used usually to point out that the noun (the noun taisetsu is referred to) should be FELT as important (precious, grave, really serious...).
And so (IF we want to find a difference between daiji/taisetsu na pointo) we can say daiji na pointo speaking about "the key-points of a reasoning", while we can say "taisetsu na pointo" if we're talking about the main points the students should (perceive as crucial and thus) remember.
However there are situations where the difference is more obvious...
taisetsu na kodomo = a beloved/dear child (precious to the speaker)
daiji na kodomo = an important child (he's the key to my 人体実験 Bwah-ah-ah!)
A context that involve such a nuance (i.e. the thing should be FELT as important) will usually be a more colloquial context, and this is the reason, imho, taisetsu can be perceived as more informal.
This doesn't mean daiji na hito is different from taisetsu na hito, if I'm talking about someone I love. But if I'm talking about the prime minister and I say "taisetsu na hito" then I'm probably implying he's precious (for my country, for instance).
This is why I said its use is narrower than the one of daiji.
Also, "daiji ni (suru/omou...)" and "taisetsu ni (suru/omou...)" are the same when they are referred to the idea of "treating something with care" (see the definition from daijirin).
Well, it should be enough, but maybe looking at kenkyuusha's examples can help and confirm what I'm saying:
〜な 〔重要な〕 important; serious; weighty; momentous; significant; material; of ⌐importance [moment, consequence]; grave; 〔貴重な〕 precious; valuable; of 《great, much》 value; valued; dear (to sb). [⇒だいじ３]
▲大切な書類 important ⌐documents [papers]
・大切な用件で on important business; on an important matter
・大切な品 something ⌐precious [that matters] 《to sb》; a ⌐valuable [precious, treasured] article; a treasure; 〈総称〉 valuables
・大切な子 a ⌐precious [beloved, dear] child; a child who is dear to sb; a child one ⌐loves [cares for, dotes on]
・大切な友[顧客] a valued ⌐friend [customer]
・命の次に大切なもの something ⌐next to [near] one's heart; a thing as precious as life itself
・この世で一番大切なもの one's most ⌐precious [important, valuable] possession (in the world); a thing ⌐that is more important to sb [(that) one cares about more] than anything else in the world
・大切である 〔重要である〕 be important; matter; be of ⌐importance [consequence]; count for a lot; 〔貴重である〕 be precious; be valuable; be of value; be a treasure 《to…》; be dear 《to…》; mean ⌐a lot [a great deal, much] to sb.
▲今日は午後から大切な会合がある. I have an important meeting this afternoon.
・これはお客様から預かった大切な品だから心して扱うように. This is ⌐something important [a valuable article] a customer left with us. Be careful with it.
・彼女はわが社にとってなくてはならない大切な人です. She is an invaluable asset to the firm. ｜ She's a person the company can't do without.
・子育てで大切なのは子育てを楽しむ心だ. The important thing in bringing up children is enjoying it.
・大切なのはこれからだ. What ⌐matters [is important] is the future.
・歯の健康には食後の歯磨きがもっとも大切です. The ⌐most important thing [first requisite] for healthy teeth is brushing after meals.
・何をするにも体が一番大切だ. In everything health should be the first consideration. ｜ Your health should be the most important thing, whatever you do.
・病気は予後が大切です. It's important to take good care of yourself during convalescence.
・この点が大切だから忘れないようにしてくれ. This part is important, so don't forget it. ｜ Bear this in mind; it's important.
大切さ importance; preciousness; value; valuableness; seriousness
・命の大切さ the ⌐preciousness [value] of life; how precious life is
・医師は命の大切さを心しなければならない. Doctors must remember how ⌐important [precious] life is.
・親子のきずなの大切さ the ⌐importance [preciousness] of parent-child bonds
・若い人は時間の大切さに気づかない. Young people don't realize ⌐how precious time is [the value of time].
大切に carefully; with (great) care.
▲子供を大切に育てる raise [bring up, rear] a child ⌐carefully [with care, lovingly]
・大切に育ててきたウサギ a rabbit (that has been) raised with great care
・大切に育ててきた菊 chrysanthemums (that) one has grown with great care
・大切に扱う handle [treat, deal with] 《an article》 ⌐with care [carefully]
・大切に保管する keep 《an article》 ⌐carefully [in a safe place]
・母が大切に取っておいた着物 a kimono my mother had carefully put aside
・大切に伝統を守る preserve a tradition carefully; guard a tradition jealously
・これは物がいいから大切に使えば長持ちするよ. It's good quality, so if you use it ⌐carefully [with care] it'll last a long time.
大切にする 〔尊重する〕 value; consider important; set a high value on…; care ⌐for [about]…; stress the importance of…; set (great) store by…; cherish; treasure; prize; make much of…; 〈物事が主語〉 be important to sb; matter to sb; be precious to sb; 〔丁寧に扱う〕 treat [deal with] carefully; look after…with care; 〔いたわる〕 take ⌐good [great] care of…; look after…; care for….
▲親を大切にする be filial (to one's parents); take good care of one's parents; look after one's parents carefully; attend to 《a bedridden parent》 devotedly; be a good ⌐son [daughter] to one's parents
・友情を大切にする treat friendship as an important matter; consider ⌐friendship [one's friends] important; friendship matters [one's friends matter] to sb
・目を大切にする take good care of one's eyes; be careful with [look after] one's eyes; 〔休ませる〕 rest one's eyes
・部下を大切にする treat one's subordinates with consideration; look after the people working for one carefully; be thoughtful about the people under one
・水[電気]を大切にする economize on ⌐water [electricity]; use ⌐water [electricity] sparingly
・ものを大切にする心 《have》 a respect for things; the attitude of using things carefully
・命よりも金を大切にする treat money as more important than ⌐life [anything else]; consider money to be ⌐more important than life [the most important thing in the world]
・本人のやる気を大切にしようじゃないか. Respect ⌐his [her] desire to try (and let ⌐him [her] do it). ｜ Give ⌐him [her] credit for wanting to try.
・自然を大切に. Preserve [Protect] nature.
・それではお体をお大切に. Good-bye! Take (good) care of [Look after] yourself.
1 〜な 〔かけがえのない・大切な〕 important; of ⌐importance [consequence]; momentous; weighty; precious; highly ⌐esteemed [prized]; valuable; a jewel of 《a son》; 《a person》 dear to one's heart.
▲母からもらった大事な品 a precious item ⌐from [given me by] my mother
・命の次に大事な物 a thing that is the most ⌐precious [important, valuable] after life (itself)
・私の大事な人 a person whom I hold dear; a person I ⌐cherish [treasure].
2 〜な 〔重要な・肝心な〕 (very) important; essential; vital; critical; crucial.
▲彼と大事な話がある. I have something important to ⌐talk over [discuss] with him.
・大事な話だからよく聞いてくれ. Listen carefully, because this is important.
・それは彼の一身上に関する大事なことだ. It is a matter of great personal importance to him.
・何よりも体が大事だ. Health is more important than anything else. ｜ Health comes before all else. ｜ Health is the first consideration.
・いいか, ここが大事な点だ. Listen carefully now, because this is the crucial point.
・それが一番大事な点だ. That's the ⌐most important [critical, essential] point.
・大事なことから先にやれ. Start with the important things. ｜ Put first things first.
・いかん, 大事な用を忘れていた. Darn! I've forgotten to do something very important.
▲大事にしまっておく treasure (up); keep in a safe place; 〔鍵のかかる所に〕 keep sth under lock and key
・お体お大事に. Take (good) care of yourself. ｜ Take care.
大事にする 〔丁寧に扱う〕 treat ⌐with care [considerately]; take good care of…; 〔大切に思う〕 prize; value; set great value on…; cherish; esteem (highly); think [make] much of…; set store by…; attach ⌐value [importance] to….
▲体を大事にする take (good) care of oneself
・目を大事にする save one's eyes
・部下を大事にする be considerate toward(s) [value] one's subordinates
・子供を大事にしすぎるからあんなにわがままになるんだ. You ⌐coddle [pamper] your children, and that's why they turned out so ⌐selfish [spoiled].
・奥さんを大事にしないと罰(ばち)が当たるよ. You'll pay dearly for it later if you don't ⌐cherish [treasure, take good care of] your wife.
・「私なんか生きる資格はない」「そんなこと言わずにもっと自分を大事にしなさい」 "I don't deserve to be alive."―"Nonsense! Stop selling yourself short."