We determine the jet vertex for Mueller-Navelet jets and forward jets in the
small-cone approximation for two particular choices of jet algoritms: the kt
algorithm and the cone algorithm. These choices are motivated by the extensive
use of such algorithms in the phenomenology of jets. The differences with the
original calculations of the small-cone jet vertex by Ivanov and Papa, which
is found to be equivalent to a formerly algorithm proposed by Furman, are
shown at both analytic and numerical level, and turn out to be sizeable.
A detailed numerical study of the error introduced by the small-cone
approximation is also presented, for various observables of phenomenological
interest. For values of the jet ``radius'' $R=0.5$, the use of the small-cone
approximation amounts to an error of about 5\% at the level of cross section,
while it reduces to less than 2\% for ratios of distributions such as those
involved in the measure of the azimuthal decorrelation of dijets.