Here is what I found when looking into this topic. I'm not a stats person so I tried to summarise how I understood it using relatively basic concepts :-)
These two frameworks treat “time” differently:
- MEM requires nested data structures (e.g. students nested within classrooms) and time is treated as an independent variable at the lowest level, and the individual on the second level
- LGM adopts a latent variable approach and incorporate time via factor loadings (this answer elaborates more on how such factor loadings, or "time scores", work).
This difference leads to different strengths of both frameworks in handling certain data. For example, in MEM framework, it is easy to add more levels (e.g. students nested in classrooms nested in schools), whilst in LGM, it is possible to model measurement error, as well as embed it in a larger path model by combining several growth curves, or by using growth factors as predictors for outcome variables.
However, recent developments have blurred differences between these frameworks, and they were termed by some researchers as the “unequal twin”. Essentially, MEM is a univariate approach, with time points treated as observations of the same variable, whereas LGM a multivariate approach, with each time point treated as a separate variable. The mean and covariance structure of the latent variables in LGM correspond to the fixed and random effects in MEM, making it possible to specify the same model using either framework with identical results.
So rather than considering LGM as a special case of MEM, I see it as a special case of factor analysis model with factor loadings fixed in such a way, that the interpretation of the latent (growth) factors is possible.
Note. If you have time-varying covariates (co-variates that change over time), there is an additional difference: Time-varying covariates have random effect coefficients for MEM (i.e. coefficients vary across individuals), but have fixed effect coefficients for LGM (i.e. coefficients vary over time).