Hi Jesse,
I would say that there is not much different on the outside. They both only work on Enterprise Geodatabases. However, the options on the AnalyzeDatasets is much more robust than the analyze, see below:
Analyze Datasets—Help | ArcGIS for Desktop
AnalyzeDatasets_management (input_database, include_system, {in_datasets}, {analyze_base}, {analyze_delta}, {analyze_archive})
Analyze—Help | ArcGIS for Desktop
Analyze_management (in_dataset, components)
I would use the Analyze Datasets on any data in the Enterprise Geodatabase for more control of what you are doing and more options.
Enterprise GIS
--- George T.