Algebra 2 courses tend to confine the number system used in class to the set of real numbers (that is, all the numbers we normally deal with in real life). Algebra 3 introduces students to a new set of numbers, called the complex numbers or imaginary numbers. The imaginary numbers are not intuitive numbers as we think of numbers normally, but this number system is useful in engineering and higher mathematics. Algebra 3's course material does not rely on complex numbers, but tends to include a fundamental introduction.
In general, Algebra 2 does not require students to understand how to read or write mathematical proofs. Algebra 2 courses that do introduce the concept of mathematical proofs tend to do so at a basic and intuitive level. Algebra 3 requires more of students in regards to the subject of mathematical proofs. Algebra 3 facilitates students in comprehending proofs and writing proofs of easily proved theorems or facts. The basic types of proofs often appear in Algebra 3 content, and students will learn when each type of proof is most appropriate for a specific problem.
It is Algebra 2 that introduces polynomials, mathematical objects that consist of high-order variables being added together. While Algebra 2 does focus on the analysis and intuition of polynomials, it usually does so at only a rudimentary level. Algebra 3 tends to teach students how to do more with polynomials, including dividing polynomials by each other, graphing polynomials and solving polynomial equations.
Most of Algebra 2 and 3 deal with equalities, which use equal signs to relate objects. Algebra 2 introduces the basic methods of solving inequalities. This is where the learning of inequalities tends to stop in Algebra 2. Algebra 3 extends on these methods, applying them to more complicated situations, such as the solving polynomial inequalities. Often students will learn how to graph these inequalities as well.