FFT and DFT are a faster way to implement convolution.
Since convolution in time domain corresponds to multiplication in the Fourier domain, you replace the sliding-window based implementation of convolution (in time domain) by a pointwise multiplication (in Fourier domain). This avoids you an extra inner loop over the input data and saves time.
With respect to boundary conditions, there is a minor difference with finite-length signals.
Using the convolution theorem to compute a convolution product in the Fourier domain implicitely assumes that the input data is periodic, i.e., when you reach the right end of the signal you re-enter by its leftmost part. This is also known as circular convolution.
If you need to have a convolution of explicitely non-periodic signals, you can add zeros at the beginning and end of your signal to avoid these border effects.