How much weight does the tripos III carry for further PhD admissions?
If you are aiming for attending a top five mathematics department in the U.S. immediately afterwards, Part III of the Tripos plays almost no role.  It's not uncommon for students to do Part III, but often after already being admitted to the U.S. department and deferring for a year.  Occasionally someone who was turned down the previous year applies again while doing Part III and is admitted, either because their application looks stronger or because they have better luck.  However, attending Part III plays little role in this.  The bar for Part III admission is lower than for top five U.S. Ph.D. programs in mathematics, so merely being admitted is not in itself a factor.  The Cambridge academic year doesn't even start until October, so there's generally no useful way to get letters of recommendation from Cambridge faculty based on Part III: they simply don't have enough to say at the point at which letters are needed.  The net effect is that you are applying with letter writers based on the previous year, and the only function of Part III is demonstrating that you are doing something productive with your time (it certainly looks better than taking a year off, for example).