BRMS vendors chiefly evolved from BRE vendors, which often added business-readable features to rule languages and evolved to decision metaphors. More recent vendors evolved directly from the need to automate decisions.
Inference rules were classically used in expert systems for advanced decision support: the rules represented the knowledge of human experts that were used in a dialog with an end-user to diagnose or advise on some situation. The decision model was a set production rules in rulesets, possibly organised via a ruleflow process, with associated data models. See also ComparedWithPRR
BRE/BRMS vendors quickly learned that rule-writing (per PRR / inference rules) was a specialist domain (cue the “knowledge engineer” or “rules engineer”) and that simpler models were required to represent decision models and units of decision making. In particular decisions could be wrapped into a decision service and used to separate concerns from processes in BPM models. Other decision automation vendors came to the conclusion that DM was more important than rule engines per se, or were important parts not so much of SOA but also EDA and Complex Event Processing.
Most Decision Management tools provide a decision table format - it constrains the domain (ie provides a business-usable template) and allows straightforward rule verification capabilities, and maps readily to a number of executable algorithms (including BRE production rules).
Decision Trees are often viewed as equivalent representations to decision tables, but with different editing and development styles.
Decision Graphs can viewed as Decision Trees that allow joins as well as branches, or as decision-focused process model elements (aka the process of making a decision).
Often complex decisions involve sequencing a collection of decision elements - such as decision tables or rules - in a kond of process. This is often called a ruleflow.
The Score Model is a common model used in scoring - a type of statistical approach, per Predictive Analytics - to allow a subjective score to be defined against some criteria for some decisioning purpose (eg credit scoring).
With the advent of Score Models and Decision Trees, that could be output from Predictive Analytics systems, links between analytics and BRMS/DM systems have been developed, chiefly using the DMG PMML standard - http://en.wikipedia.org/wiki/PMML