Points to look at before we analyze the answers.
A) Presence of contrast word "Although" at the beginning of the sentence
2) Presence of multiple subjects - Mathematical Physics and Physical Mathematics.
Option A: Although mathematical physics are not the same as physical mathematics, the
The Subject Verb agreement does not agree her. The subject being 'Mathematical Physics' and the verb 'are' are singular and plural respectively.
Option B: Mathematical physics and physical mathematics are not the same, but the
The Subject and Verb are in agreement- with now multiple subjects having been added- Mathematical physics and physical mathematics with the verb 'are'
Option C: While physical mathematics are not the same as mathematical physics, the
Same probelm- subject verb agreement.
Also note 'Although' is removed , but that does not make any change to the sentence.
Option D: Mathematical physics is not the same as physical mathematics, the
This uses a singular verb- 'is'.
The Subject and Verb agree here- however if you look at the contrast word at the beginning of the sentence 'Although' has been removed.
This makes both parts of the sentences an independant clause.
And independant clauses cannot be linked by a comma- this error is called
comma splice.
Option E: Mathematical physics is not the same as physical mathematics, and
Similar as Option D - but the word at the end of the sentence changes to 'and'.
This also causes a change in the meaning.
Option B is the correct answer.