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Maximum likelihood estimation of the Bingham distribution is difficult because the density function contains a normalization constant that cannot be computed in closed form. Given the availability of sufficient statistics, Approximate Maximum Likelihood Estimation (AMLE) is an appealing method that allows one to bypass the evaluation of the likelihood function. The impact of the input parameters of the AMLE algorithm is investigated and some methods for choosing their numerical values are suggested. Moreover, AMLE is compared to the standard approach which numerically maximizes the (approximate) likelihood obtained with the normalization constant estimated via the Holonomic Gradient Method (HGM). For the Bingham distribution on the sphere, simulation experiments and real-data applications produce similar outcomes for both methods. On the other hand, AMLE outperforms HGM when the dimension increases. [ABSTRACT FROM AUTHOR]