Question

What’s the difference between these two postpositions which can both be translated as “other than, besides, in addition to”?

के अलावा
के अतिरिक्त

Is के अतिरिक्त a “शुद्ध” phrase?

Answer

There isn’t any difference in meaning; just as English, Hindi has synonyms, and the choice among the synonyms is a matter of style, but not strictly of meaning. As you suggested, के अतिरिक्त is a more formal postposition.

Note: “शुद्ध” means “pure”; it is a term that Hindi speakers use to describe very formal words, often those words that come from Sanskrit.

के अलावा is more common. These phrases are basically synonymous,
just as “besides” and “other than” in English. Note that it means “in addition to”; for the meaning “instead of”, के बजाये could be used.
तुम्हारे अलावा कौन जानेवाले हैं – “Besides you, who else is going to go?”
क्या आपको हिंदी पढ़ने के अलावा कोई शौक है – “Do you have any hobbies other than studying Hindi?”
वह काम के अतिरिक्त कुछ नहीं करता – “He doesn’t do anything besides work” |