why is it that the amount when calculating the compound interest manually different from when I use the compounding formula A=p(1+i)n?

You haven't really explained how you concluded that your formula is giving a different result than whatever result somebody else is getting.

Such calculations are obviously subject to small discrepancies due to rounding, but aside from that, the obvious answer is that somebody is doing the arithmetic wrong.

Are you sure you're applying the formula correctly? Consider a debt with 1.5% monthly interest, or a nominal interest rate of 18%. With interest calculated on a monthly basis, the effective interest rate is determined by calculating (1.15^12) = 1.1956, then subtracting one for an effective rate of 19.56%.

Incidentally, more frequent compounding has a fairly limited effect on things. For instance, daily compounding would result in an effective rate of 19.716%, while with continuous compounding, the effective rate is 19.7217%.