These phrases limit things as exceptions in their own unique ways.
After a noun phrase of some sort, ～除いて is used to mean “except…” and is often left in ひらがな. It is a general phrase and will differ with the emotive power of the following grammar point ～をおいて. The verb 除く can mean “to remove/exclude”, and “exclude” can also be expressed with the verb 除外する, which is of the vein of “to set aside/rule out”.
Except for one vessel, all of the carriers sank.
Yesterday, except it being hot, we had a good time.
Please include me out.
I believe it in the equality of all people except reporters and politicians.
The weather was good except in the north.
Except me, it seems that everyone knew about it.
It was decided that women were to be taken out of the inquiry subjects.
Variant Note: In more formal literary fashion, this can also be seen as ～のぞき.
In regards to the all-round band lift of the sale of over-the-counter drugs using the internet that a government regulation reform meeting is seeking, Chief Cabinet Secretary Kan, Prime Minister Tamura of the Ministry of Health, Labor and Welfare, and four other bureaucrats in connection are forecast to further the coordination in the direction of lifting the ban in large part with exception to a small amount of over-the-counter drugs of which the evaluation of side-effect risks is not over.
From NHK on June 4, 2013.
Warning Note: Do not confuse this with the verb 覗く, which means to “to peep/take a look at” and has the same pronunciation.
～をのぞいて VS ～以外
This is very similar to ～以外. One thing to notice, though, is that ～以外 is a suffix, and as seen below, the phrase is more affirmative and somewhat more formal sounding.
With exception of Kyushu, earthquakes occurred everywhere.
There is no method other than this.
～をおいて means "apart from". This phrase is used to highly evaluate something or someone. So, it's a good thing. There is always something like ほかに...（い）ない after it. When you are generally saying "aside from", use ～を除いて. Like ～除いて， ～をおいて is used after noun phrases.
This phrase originally showed a meaning of "without emphasizing A as something important, one treats A as being useless and throws it out/places it to the side/separates it apart/removes it. In this sense, Aをおいてほかに（い）ない becomes a double negative expression, which then makes it an extremely powerful affirmative statement of A being number one.
In Japanese it is usually the case that very powerful expressions are kept in the written language, and although this is the case for をおいて, in rather formal situations, it can be used to declare one's top recommendation. It doesn't necessarily have to be in formal situations; however, as the first example sentence demonstrates clearly.
Apart from you there's no responsible person!
There is no one to rely apart from her.
There was no way but this.
Aside from that
Apart from Kyushu and Shikoku, no matter where there are many deaths and major catastrophes have occurred in the Tohoku-Kanto Earthquake Catastrophe.
～別 VS ～をおいて
The following paraphrases with 別 is used rather than ～をおいて because the situation is not one of pointing out as the best but something that should be treated separately.
16a. 震度7 は別として、釜石市には津波が押し寄せました。
Apart from a 7 earthquake, Kamaishi City got flooded by a tsunami.
Word Note: The 震度 system is on a different scale than the Richter scale.
～ならでは is one of those instances where classical grammar holds on strong. The なら in this expression, like the particle なら comes from the 未然形 of the Classical Japanese copular verb なり. It is then followed by a classical usage of で, which is not related to the contraction of にて. In this case, it is the contraction of ～ずて, which is equivalent to ～なくて. は, here, is here for contrastive purposes. If the pattern were translated into something solely Modern Japanese, it would be equivalent to ～でなくては・～でなければ.
This pattern is used to express the brilliance/wonderfulness of something by claiming that only it is as such. In the unabbreviated state of a sentence with it, a complementing verbal/adjectival phrase follows, but this can be omitted out if that phrase is being used as an attribute by replacing it with の, giving ～ならではの, which is the more common form of the pattern.
The entire phrase, which is “AならではBない（C ）” is, then, equivalent to expressions such as “AであってはじめてB” and “AだからこそB”. In the case that A is a commonplace noun, ～ならでは is of the sense of the former coming from the position of societal wisdom/common sense. However, when A is particular, it shows that only A can do C.
Without you, this cannot be done!
That is only a matter of thought possibly by a person from China.
That is charm only that of Osaka Dialect, isn’t it?
This can only be a conception of the company president, isn't it?
This is a characteristic only in this textbook.
The goodness of this flavor that only an old shop for 50 years can offer!
This experience can only be in Okinawa!
Grammar Note: As you can see, another exceptional thing about ～ならでは is that it can be followed by ～です with the rest of the pattern omitted.
This is as expected flavor that can only be from a first rate restaurant chef.
Phrase Note: さすが, which is an adverb that shows something as something to be expected, is frequently used with ～ならでは.