What is the difference between pure and applied math?

**Pure Mathematics**
Pure Mathematics courses emphasize the specialized language of algebra as the preferred method for learning mathematical concepts and for solving problems. Students learn about mathematical theories, find exact value solutions to equations, and use formal mathematical reasoning and models in problem solving.

Students in both Applied Mathematics and Pure Mathematics programs are expected to graph and solve algebraic equations, and to solve problems in trigonometry, finance, statistics and probability. In both programs, students use the graphing calculator to visualize and identify patterns and relationships in algebraic equations.

**Applied Mathematics**
Think of applied mathematics as the practical and experiential part of the equation resulting from the reasoning and inspiration of pure mathematics.

Applied mathematics or computational sciences encompasses the many analytical and numerical methods used to solve scientific problems. It is at the core of many disciplines ranging from engineering and science to business and economics. It is essential to be able to apply many different mathematical techniques and to handle problems involving statistical data. It is also necessary to be able to take a practical problem, from engineering for example, and turn it into a mathematical problem; this is referred to as mathematical modeling.

Applied mathematics includes the area of scientific computing, which involves many components of computational processes - numerical analysis, algorithms for machines with vector and parallel architectures, visualization, simulation and computer-aided design.

Applied mathematicians use equations and computer simulations to:

design computer chips;

set airfare and insurance rates;

model electric power grids;

study genetic engineering;

create new automobile and airplane design;

study global climate change.

It is mathematics used to model and solve practical, real-world problems.

Source:

http://www.bukisa.com/articles/213890_differences-between-applied-mathematics-and-pure-mathematics
Which would be more useful as a quant?

Both programs help students develop the critical skill of using mathematics to find solutions to problems involving real-life situations.

What are the best colleges for applied math?

Tons of colleges! Better to concentrate on more important criteria for selection like - direction(profession) to follow, tuition variation, employment perspective, usefulness for preparation for more advanced academic courses(if intending...), etc.

Finally, is programming a huge part of being a quant or can I going along having minimal programming skills?

It's true that it depends on what you want to do but in most cases the answer is YES - it is an integral part of being a quant.(at least enormously helpful). While you can find some jobs purely academic rather than practical, unless you are well-experienced professional holding many university degrees, it is expected that your employer expects practical skills from you. You can find many discussions about the role of programming on the forum.

Hope that helps.