Difference Between Convolution and Multiplication
Multiplication in your first sentence is term-by-term multiplication: $z[n] = x[n]y[n]$ for all $n$. Convolution, for discrete-time sequences, is equivalent
to polynomial multiplication which is not the same as the term-by-term
multiplication. Convolution also requires a lot more calculation: typically $N^2$
multiplications for sequences of length $N$ instead of the $N$ multiplications
of the term-by-term multiplication.

The key point of Fourier analysis is that term-by-term multiplication in one
domain is the same as convolution in the other domain.  So, in order
to calculate the results of a convolution, you can either do it directly,
using $N^2$ multiplications, or transform to the other domain, do a term-by-term
multiplication, and transform back. This requires two transformations
to go from one domain to the other, $N$ multiplications in the other domain,
and one inverse transformation to come back to the domain
where the convolution result is needed. This more complicated process
can, in fact, require less computation because the transformations can
be done very efficiently via the Fast Fourier Transform (FFT) algorithm
which requires about $N \log N$ multiplications. So, compare the computational
effort in calculating the three transforms and doing the term-by-term
multiplication
to $N^2$ to see if the FFT gives you a more efficient method of computing
a convolution.