In this line:
он сел наземь между мною и Верой
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Both forms stand for Instrumental case of the 1st person singular pronoun я.
The form мною is an older variant of мной, it fades away now as the second option becomes more popular. You can see the distribution of the two forms over years here.
Reference service of gramota.ru sometimes calls both variants equivalent and sometimes recommends to use мной, because мною is considered obsolete.
Another reason why мною may be used is that in poetic texts the author wants to keep the correct metre or rhyme (but also for some other, maybe personal reasons). For example, Pushkin usually uses мной, but in the following poem he writes мною:
Алина! сжальтесь надо мною.
Не смею требовать любви.
Быть может, за грехи мои,
Мой ангел, я любви не стою!
Finally, the form мною can be used in modern texts that for stylistic reasons have to look like old texts.
I don't know what is the exact reason why it is used in your phrase. Maybe it is an text from 18th century.
The modern form мной (or any other sg. inst. ending on
-ей) is quite a recent apocope of мною which had been a norm till the late XVIII.
You may observe it on this graph.
Russian tends to reduce ending vowels in weak position which can also be observed in particles like -бы/-б, -же/-ж, -ли/-ль etc. (reflected by the orthography) and also -то/-т, -ка/-к etc. (which are not yet reflected). The difference between the sg. inst. in declension I is a part of the same process.
Currently, the latter form is considered archaic, but may be still used in poetic, high style speech or just for euphonic purposes. The two forms are rhythmically different so interchanging them can help in composing a verse.
In your case, it is most probably used to avoid й-и cluster which is somewhat hard to pronounce without omitting the й.