To my ears, '자' and '쟈' seem to be pronounced the same. Are they pronounced differently, and if so, what do you perceive to be the difference?
They are not distinguished properly in modern Korean. It includes 자/쟈, 저/져, 주/쥬, 조/죠, 제/졔, 재/쟤 and ㅊ and ㅉ-equivalents too. This is because ㅈ(/t͡ɕ/) is already palatalized in modern Korean, or at least in the Standard Korean pronunciation, so that adding /j/ after it doesn't make a difference. That's why there are no words spelt with 쟈/져/쥬/죠/졔/쟤 in Korean vocabulary. The standard orthography forbids it. Not even loanwords. Loanwords spelt with 쟈/져 etc are wrong according to the standard and should be corrected.
Except when this spelling is orthographically necessary:
가져 < 가지어
다쳐 < 다치어
The only reason there's a 져 and 쳐 in there is to show that they are shortened forms of 지어 and 치어. They are pronounced same as /가저/ and /다처/.
Doubtful natives, read this for more info.
Are they pronounced differently?
Yes, they are pronounced differently, but they sound very similar. It will be very difficult to distinguish them. The vowel 'ㅑ [yah]' is a cousin of 'ㅏ [ah]' and you generally pronounce 'ㅑ' a little longer than 'ㅏ'.
You can feel the difference if you read the below very slowly and loudly.
아 야 어 여 오 요 우 유 으 이
가 갸 거 겨 고 교 구 규 그 기
자 쟈 저 져 조 죠 주 쥬 즈 지
The '쥬' in 쥬얼리 is definitely different from 주. Saying "They are not distinguished properly in modern Korean." is wrong. I would rather say "They sound very similar but the pronunciation is distinguishable."
Just because "The standard orthography forbids it" doesn't necessarily mean the pronunciation doesn't exist. It does exist however minimal the difference is
What do you perceive to be the difference?
You can tell the difference by only distinguishing [yah] sound from [ah] sound .
There is no Korean word that starts with '쟈' and 네이버 국어사전 link shows only 4 words, two of which are used in North Korea and the other two are used only in Jeju (제주도) dialect.
Unless someone wants their brand or product name to look very unique and peculiar, there is no chance that '쟈' will be used in Korean. You can write '자' when you hear '자' or '쟈'.