The biggest difference between the mRNA type and the viral vector type is how the genetic code for the spike protein is delivered.
The adenovirus injects DNA into cells, which is then transcribed in to RNA, which is then translated in to protein by the normal cellular process. The protein is what causes the immune response, giving us immunity.
The mRNA type vaccines are delivered with nano sized lipid spheres, very much like soap bubbles, which simply merge with the cell membrane. The mRNA is then immediately translated in to protein. This technology cuts out quite a few steps. The mRNA itself is quite unstable (hence the need for -80 C freezers) and doesn't last very long in the cell. It also never goes in to the nucleus so there is no way it can alter the DNA of the cell.
Quite the elegant technology, and in the real world, with millions of doses administered, there have been zero safety issues.