They both have same qualification , so I don't think it matters for a long term.
Yeah, if it comes to job offers from corporate hospitals then pay-scale is decided on the basis of interview. And during this, they may consider your background . But it matters only in Engg. Interviews that most.
AIIMS tag gives you a head start in whichever stream you select, but later on its individual calibre that matters.
It just helps in head start. After all what matters is knowledge, skills and confidence.
Hope it helps.
Good luck....

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