In Procedural Content Generation (PCG) in computer games, you can generate procedurally a variety of contents, from characters, weapons, songs, maps, dungeons, story, and even entire NPCs. In few words, you start from a set of base images and rules, you then apply these rules on a random subset of images (note that the rules are usually based on some random generator, so there is variability and perturbations in them too - but this depends mostly on the role of PCG and what type of content you want to design.) to create new images. You can repeat this process at each gameplay, for example, to generate an entire new experience for the player or use the method to generate candidates, which are then refined by hand - so picking the very best generated ones.
In contrast, generative AI (or deep generative models) starts by training a (usually large) model on a (large) dataset, such that when noise is injected (or, as in some other approach, the noise is the input) you can generate (i.e., synthesize) a new sample that looks like a plausible image from that dataset. In this regard, there are multiple effective methods like generative adversarial networks and diffusion models, but in some cases also variational autoencoders and normalizing flows are fine.
The difference between PCG and GenAI is that, in the former case you design by hand the base images and rules, whereas in the second case you design the model and, in some cases, also curate the dataset. So, in PCG the generation is controlled by your rules, whereas in GenAI is determined by what the model has successfully learned.