Separable nonlinear least squares (SNLLS) problems have attracted interest in a wide range of research fields such as machine learning, computer vision, and signal processing. During the past few decades, several algorithms, including the joint optimization algorithm, alternated least squares (ALS) algorithm, embedded point iterations (EPI) algorithm, and variable projection (VP) algorithms, have been employed for solving SNLLS problems in the literature. The VP approach has been proven to be quite valuable for SNLLS problems and the EPI method has been successful in solving many computer vision tasks. However, no clear explanations about the intrinsic relationships of these algorithms have been provided in the literature. In this paper, we give some insights into these algorithms for SNLLS problems. We derive the relationships among different forms of the VP algorithms, EPI algorithm and ALS algorithm. In addition, the convergence and robustness of some algorithms are investigated. Moreover, the analysis of the VP algorithm generates a negative answer to Kaufman's conjecture. Numerical experiments on the image restoration task, fitting the time series data using the radial basis function network based autoregressive (RBF-AR) model, and bundle adjustment are given to compare the performance of different algorithms.