I have no experience with Mac, InDesign, or the Tamil language, but I won't let that stop me from taking a stab in the dark! Regarding the PDF that passed for you but failed for the client - was the font embedded? I am not really sure if that would make a difference but it is something I would check. You can embed fonts in Acrobat Pro using Preflight - at least on a PC - unless the font forbids embedding. I would think if the font is embedded it should behave the same on any machine, but again, not completely sure.
Thank you for your reply. The fonts, no matter what fonts I am using, are embedded in PDF. The issue is not that is showing differently on a computer versus another. It is simply something related to font coding. As I read on a forum, it looks like Arial Unicode is showing Old Tamil letter/signs.
To fix the issue with the vowel sounds you just have to select the text and enable the World Ready Para Composer option in the Paragraph panel. Hope this helps!