They sort-of appear in both, but in the original [1] they're explicitly mentioned and used to build tension, whereas in the remake [2] they're not mentioned explicitly, but the engine is started with one.
According to the Wikipedia article linked above, it was included in the remake as well (I haven't seen it):
> The starter became famous as a plot device in the 1965 movie The Flight of the Phoenix, in which pilot Frank Towns (James Stewart) has a limited number of cartridges with which to start the makeshift aircraft's engine; this was also featured in the 2004 remake of the film.
The original, not the remake.