Face recognition systems have significantly advanced in recent years, driven by the availability of large-scale datasets. However, several issues have recently came up, including privacy concerns that have led to the discontinuation of well-established public datasets. Synthetic datasets have emerged as a solution, even though current synthesis methods present other drawbacks such as limited intra-class variations, lack of realism, and unfair representation of demographic groups. This study introduces GANDiffFace, a novel framework for the generation of synthetic datasets for face recognition that combines the power of Generative Adversarial Networks (GANs) and Diffusion models to overcome the limitations of existing synthetic datasets. In GANDiffFace, we first propose the use of GANs to synthesize highly realistic identities and meet target demographic distributions. Subsequently, we fine-tune Diffusion models with the images generated with GANs, synthesizing multiple images of the same identity with a variety of accessories, poses, expressions, and contexts. We generate multiple synthetic datasets by changing GANDiffFace settings, and compare their mated and non-mated score distributions with the distributions provided by popular real-world datasets for face recognition, i.e. VGG2 and IJB-C. Our results show the feasibility of the proposed GANDiffFace, in particular the use of Diffusion models to enhance the (limited) intra-class variations provided by GANs towards the level of real-world datasets.