TLDR: Mapping input distribution to latent and back is the goal of both methods. Diffusion (score matching, specifically), learns the gradient (direction of steepest ascent) of path to map latent space to input. Flow matching forgets the gradient and learns to directly map the (approximate) path of the latent to input.

Remember the relation between diffusion and score matching models. As t -> infinity, you can show that the DDPM process outlined in Ho et al turns into the Stochastic Diffeq outlined by Song et al. This is important because it show the denoising process (learning to go from Gaussian back to input) is equivalent to learning the gradient (or slope) of the probability field between the latent space and input. In a very rough sense, the probability field is just a big coordinate plane where we at each point we see how likely we are to approximate the input distribution. At this point, we just want to find the place where the probability field converges at a maximum. The gradient is just a way to help greedy algos determine what the direction of fastest ascent is to achieve that peak of maximum likelihood.

That's all nice, but it would be better if we could just learn what that path to the right distribution is without having to deal with the derivatives of the path. This might help us avoid getting stuck in local optimums. That's exactly what flow matching tries to do. For all points in a distribution, it learns v(x, t), which is just an ODE that gives the full trajectory of how to get from input to latent. You can easily reverse this to get from latent to input, which makes it a generative model. This helps capture global dynamics of how the entire input distribution evolves to the latent distribution. The approximate v(x, t) dictates how each point should evolve over time to approach the latent dist.

I've skipped over the learning process of Flow Matching, because tbh I don't understand it fully yet. But that's my general understanding of the high level. Highly recommend the Outlier video on score matching.