When we refer to the "Rosette API", we're talking about a typical SaaS cloud-based web API. The Rosette API supports the majority of functions of the SDK -- the API is a subset of the SDK in most regards. The major features the SDK has that the API lacks are stateful and user-customized features - mainly Name Indexes (to disambiguate against a custom curated list of names, rather than the simple pairwise matching the API is limited to) and model training -- our entity extractor can be trained against a dataset/model of your own in the SDK, but not in the API. However, the API contains Sentiment and Categorization, which are not fully featured SDKs just yet.
A newly-available product, really a "third flavor" of our text analysis suite, is "Rosette Enterprise". Users can run their own web server with all the same functions as the Rosette API, but in an on premise model (which solves cloud security worries), and customize (models/indexes) as needed.
Rosette Enterprise can be easily customized to fit into the system you need on-premise -- please ask us if you have any questions. We have an experienced team ready to help integrate Rosette Enterprise into your envirornment.