What is the difference between 差点儿 and 几乎？

it's better write down the sentence. depends on the concrete situation.  

意思用法差不多。差点儿有一种遗憾的语气在里面吧。

Well,from the prospective of spoken chinese,NO big difference.And 几乎  is more kinda of being literary.

This is a really great big question, I had to use other references to clarify the differences. Among them, this word of 差点儿 is not easy to understand.

1, 差不多 (adj.)

2, 差（一）点儿 (adv.)

3, 几乎 (adv.)

The three above are all means average, almost, nearly.

1, 差不多 (adj.)

(1) 差不多 means "about the same in level, state, time, quantity, etc." and can be an attributive, adverbial, predicate, etc.:

① 这两间房子差不多大，都是40平方米左右。

The areas of the two rooms are nearly the same. Both are about 40 square meters large.

② 差不多大的成年人都可以拿 得动这样的重物。

The grownups of that age should be able to cany such heavy thing.

③ 这两件上衣的样子差不多。

The designs of the two coats are very alike.

差（一）点儿 and 几乎 cannot be so used.

(2), 差不多 means "nearly", used as an adverbial:

④ 他的左眼差不多完全失明了。

His left eye is almost blind completely.

⑤ 儿子结婚请客，他差不多把全 村的人都请到了。

He invited nearly all the villagers to celebrate his son's wedding.

⑥ 他差不多干了20年的记者。

He has been a journalist for about 20 years.

几乎 can also be used in examples ④，⑤，and ⑥，but with more stress and there is a literary flavor. In everyday speech, 差 不多 is mostly used:

⑦ 这个西红柿真大，差不多有一斤。

The tomato is very big. It is about one jin.

2. 差（一）点儿

(1) Indicating that one feels lucky because some unwished-for state of affairs was not, though, nearly realized. It is very colloquial, e.g.:

① 老大妈脚下一滑，差（一）点儿跌倒。

The old grandma slipped and nearly fell over.

② 这个棋子差（一）点儿被他吃掉。

The piece of mine slightly escaped being taken by his.

③ 这场大病，他差（一）点儿丢了性命。

This serious illness nearly sent him to heaven.

What is noteworthy is that the unwished-for state of affairs can be expressed in the negative form without changing the meaning, and it is even more colloquial: example ① can be changed into 差点儿没跌倒, example ② into 差点儿没被他吃掉, example ③ into 差点儿没丢了性命. The three examples above can also use 几 乎 instead.

Sometimes this way of expression is not used to indicate the feeling of rejoicing, but some extreme cases (mostly used with words of the extreme meaning):

④ 她气得差（一）点儿哭了出来。

She was on the verge of tears because of anger.

⑤ 闻到那股味道，我差（一）点儿吐 出来。

I was on the verge of vomiting at the smell.

几乎 can also is so used.

(2) A wished-for state that seemed not to be realized has actually realized after all (used in the negative form):

⑥ 我买的那本书是最后一本，我差（一）点儿没买到。（买到了）

The book I bought was the last one. I almost missed it. （I got it）

⑦ 这次考试，题目太多了，他差（一）点儿没答完。（答完了）

There were too many questions in the test. He nearly failed to finish it. (He finished it)

⑧ 因为机票难订，她差（一）点儿没走成，幸亏最后买到了退票。 (走成了)

As it was hard to get an air ticket, she almost failed to leave. Fortunately, she got a returned ticket. (She left)

几乎 can be so used too.

(3) Some state of affairs which seems to be realized but actually not. Here is a flavor of regret (就 is often used):

⑨ 我差（一）点儿就要买到音乐会 的票了，谁想到前面那个人一下 子买了 15张。(没买到)

I nearly bought the ticket for the concert. Unfortunately, the person prior to me bought 15 tickets. (I did not get one)

⑩ 他差一点儿就得冠军了，可惜最后几秒时被别人超过了。（没得上）

He nearly win first place; however, he was exceeded in the last few seconds. (He did not win it)

几乎 can be so used too.

In the examples in (1) (2) (3), all those in which 几乎 can be used instead would not be so colloquial as using 差(一)点儿.

3. 几乎

(1) Indicating "estimation, almost, barely":

① 几年不见,他变得我几乎都不敢认了。

I nearly could not recognize him for he changed greatly these years.

② 这个公司，博士几乎占了1/3。

Doctors in this working unit nearly account for one-third.

差不多 can also be used instead but sounds less forceful.

(2) The usage is the same as "差(一)点儿" (1) (2) (3), but not so colloquial as it:

③ 老大妈脚下一滑，几乎跌倒。 （没跌倒）

The old grandma slipped and nearly fell over, (she did not fall over)

④ 因为考题太多，他几乎没答 完。（答完了）

There are too many questions in the test. He almost failed to finish it. (He finished it)

⑤ 他几乎得冠军了。（没得到）

He nearly won first place. (He did not get it)

(3) The usage is the same as 差不多 (2), but the level, state, or quantity must be of an extreme or exaggerated nature:

⑥ 他的左眼几乎完全失明了。

His left eye is almost blind completely.

⑦ 他的病几乎全好了。

He almost got over it.

Question: Did he get into Peking University?

他差点儿就考上北京大学了。

他差点儿没考上北京大学了。