We investigate the ultrafast electron dynamics triggered by terahertz and optical pulses in thin platinum and gold films by probing their transient optical reflectivity. The response of the platinum film to an intense terahertz pulse is similar to the optically-induced dynamics of both films and can be described by a two-temperature model. Surprisingly, gold can exhibit a much smaller terahertz pulse-induced reflectivity change and with opposite sign. For platinum, we estimate a 20% larger electron-phonon coupling for the terahertz-driven dynamics compared to the optically-induced one, which we ascribe to an additional nonthermal electron-phonon coupling contribution. We explain the remarkable response of gold to terahertz radiation with the field emission of electrons due the Fowler-Nordheim tunneling process, in samples with thickness below the structural percolation threshold where near-field enhancement is possible. Our results provide a fundamental insight into the ultrafast processes relevant to modern electro- and magneto-optical applications.