I agree with the comments made above by @Patti Lee and @bfoste01, though I believe this answer can add some clarity.
Both latent class analysis (LCA) and the Rasch model are models for categorical data (namely binary and ordinal data) that posit the existence of one or more (potentially related) latent variable(s). The core difference between the two methods is that LCA assumes the presence of a categorical latent variable(s). In contrast, the Rasch model assumes the presence of a continuous latent variable(s).
For example, say you have five binary (0/1) items designed to measure English proficiency to classify a student as either proficient or not proficient. Suppose you were to apply LCA to this data set. In that case, students could only be classified as proficient or not proficient (you would also have access to each student's probability of being either proficient or not proficient). If the Rasch model were to be applied, students would receive a score corresponding to their sum score (i.e., a score corresponding to 0, 1, 2, 3, 4, & 5). Then, if you want to use these scores to classify someone as proficient or not proficient, a cut point along 0 - 5 would need to be determined to reduce the six possible (continuous) scores into a binary classification.
The above example illustrates a scenario where either LCA or the Rasch model could be plausibly considered; however, not all applications of LCA are also appropriate for the Rasch model (and vice versa). For example, you discuss scale development, and LCA is typically not used for this purpose. However, the Rasch model is since psychometric scales are typically conceived to be continuous when the focus is to establish a scale to measure latent phenomena (though there are notable exceptions to this, for example, diagnostic classification model (DCM)s).
Finally, see Borsboom et al., 2016 for an accessible discussion of the differences between latent variable models for continuous (e.g., the Rasch model item response theory (IRT) models, and factor analysis (FA) models) and categorical (e.g., LCA, latent profile analysis (LPA), and DCMs) traits.
Borsboom, D., Rhemtulla, M., Cramer, A. O., van der Maas, H. L., Scheffer, M., & Dolan, C. V. (2016). Kinds versus continua: A review of psychometric approaches to uncover the structure of psychiatric constructs. Psychological medicine, 46(8), 1567-1579.