The two approaches are conceptually very similar and often lead to similar results in practice. The key difference is that latent class analysis (LCA) treats people within a given latent class as "statistically identical" (all individuals in a given latent class are assumed to have the exact same conditional response probabilities), whereas in the mixed Rasch model (MRM), individuals can differ within a given class (because a Rasch/dimensional model holds within each class). The MRM is thus a hybrid between a latent class and a dimensional (latent trait/factor) model. LCA is a special case that follows from the MRM when individuals do not differ within classes. I believe the original paper by Juergen Rost addresses the similarities and differences between LCA and the MRM at least to a certain extent:

Rost, J. (1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement, 14(3), 271-282.

https://conservancy.umn.edu/bitstream/handle/11299/113859/v14n3p271.pdf?sequence=1

Christian Geiser thank you for your responce and I will read the recomended paper.