The scope of analysis is far greater than the scope of algebra. Analysis applies to much more than math and even economics. Analysis is not even limited to quantitative analysis involving math; it can be the observation of nearly anything. When looked at broadly, algebra is a tool for observing nearly anything, but algebra has many characteristics unique to its pursuit that may or may not exist in broader analysis.
Modeling is prevalent in analysis. But not all analysis involves modeling. Algebra, however, requires the use of quantitative models that frequently reference other things. A common characteristic of algebra is the use of variables, usually letters, that stand in for something else, often numeric values. For a simple algebraic expression (e.g., a function of x) set equal to a constant, the variable x always has the same value. By changing what the algebraic expression is set equal to (assigning a value to y) or making the expression itself more complex (among other things), one can model relationships. An assigned value for x, y, or some other variable for an algebraic expression is often a quantitative measure of a real-life object or phenomenon, such as in word problems that ask how fast a train is traveling.
One a basic level, you use mathematics -- frequently algebra -- to solve problems. Whether you are a carpenter or a banker, you likely use algebra. As used in day-to-day life, the objective of algebra is to solve problems or to produce a value or answer. Analysis, on the other hand, does not necessarily seek specific answers. Practitioners generally use analysis to obtain broader understanding. Thus, they often engage in more pure modeling than problem solving. As algebra and other branches of mathematics advance, they move toward modeling, where solutions to problems are accurate models of a real-world scenario, not a specific value.
For all the areas where the two overlap, perhaps the biggest departure is the area of non-mathematical analysis. For example, you can analyze poetry or love without using mathematics at all. Such analyses need not even be logical (e.g., employ the true/false logic often found in math). Much analysis results in nonquantifiable understanding.