Latent Class Analysis is in fact an Finite Mixture Model (see here). The main difference between FMM and other clustering algorithms is that FMM's offer you a "model-based clustering" approach that derives clusters using a probabilistic model that describes distribution of your data. So instead of finding clusters with some arbitrary chosen distance measure, you use a model that describes distribution of your data and based on this model you assess probabilities that certain cases are members of certain latent classes. So you could say that it is a top-down approach (you start with describing distribution of your data) while other clustering algorithms are rather bottom-up approaches (you find similarities between cases).
Because you use a statistical model for your data model selection and assessing goodness of fit are possible - contrary to clustering. Also, if you assume that there is some process or "latent structure" that underlies structure of your data then FMM's seem to be a appropriate choice since they enable you to model the latent structure behind your data (rather then just looking for similarities).
Other difference is that FMM's are more flexible than clustering. Clustering algorithms just do clustering, while there are FMM- and LCA-based models that
- enable you to do confirmatory, between-groups analysis,
- combine Item Response Theory (and other) models with LCA,
- include covariates to predict individuals' latent class membership,
- and/or even within-cluster regression models in latent-class regression,
- enable you to model changes over time in structure of your data etc.
For more examples see:
Hagenaars J.A. & McCutcheon, A.L. (2009). Applied Latent Class
Analysis. Cambridge University Press.
and the documentation of flexmix and poLCA packages in R, including the following papers:
Linzer, D. A., & Lewis, J. B. (2011). poLCA: An R package for
polytomous variable latent class analysis. Journal of Statistical
Software, 42(10), 1-29.
Leisch, F. (2004). Flexmix: A general framework for finite mixture
models and latent glass regression in R. Journal of Statistical
Software, 11(8), 1-18.
Grün, B., & Leisch, F. (2008). FlexMix version 2: finite mixtures with
concomitant variables and varying and constant parameters. Journal of
Statistical Software, 28(4), 1-35.