As the title asks--what is meant by "Theological Interpretation of Scripture", a.k.a. TIS, and how does it differ from other methods?
I think examples of types of interpretation answer the question best.
Literary Interpretation – 'How does this text fit within the text before and after?' 'What kind of language is used, literal, symbolic? This is used for all literature, spiritual or not.
Historical Interpretation – 'What did this mean to those people when they wrote it, and how was it understood by them?' For example, how was Genesis understood before there was a King in Israel? How does history explain the context?
Theological Interpretation – 'How does this passage manifest the mind of God?' 'How does this verse fit into the central theme of scripture, i.e. the Christ, his death and resurrection?'
Biblical Theological Interpretation, or Biblical Theology asks - 'How does this verse fit into the gradual revelation of God's redemptive plan?' It is a single question, meant to restrain your view. This is therefore one subset of a general 'Theological Interpretation'.
Theological Interpretation of Scripture (TIS) is a new 'buzzword' that seems to support the above forms of interpretation as a collection. According to Gregg R. Allison, in Theological Interpretation of Scripture: An Introduction and Preliminary Evaluation:
Due to its newness, TIS continues to be rather difficult to define, and while no consensus definition exists, I offer the following: TIS is a family of interpretive approaches that privileges theological readings of the Bible in due recognition of the theological nature of Scripture, its ultimate theological message, and/or the theological interests of its readers.
I think a key observation by Gregg Allison gives insight into the 'spirit' of the 'movement':
TIS is commonly viewed as a self-conscious effort to take back the interpretation of Scripture from the academy and (re)situate this endeavor in the church.
So what we are talking about here seems to be letting theologically mature people answer tough questions from the Bible, without being overly muted by highly academic and potentially atheistic literary criticism, which does not tend to allow '3. Theological Interpretations'. At the same time TIS requires proper forms of exegesis. TIS therefore seems to be a movement that encourages things like...well...this website.