As I understand it, since a solution to a linear program always occurs at a vertex of its polyhedral feasible set (if a solution exists and the optimal objective function value is bounded from below, assuming a minimization problem), how can a search through the interior of the feasible region be better?  Does it converge faster?  Under what circumstances would it be advantageous to use simplex method over interior point methods?  Is one easier to implement in a code than the other?
What are the advantages/disadvantages of interior point methods over simplex method for linear optimization?
convex optimization;linear programming
Based on personal experience, I'd say that simplex methods are marginally easier to understand how to implement than interior point methods, based on personal experience from implementing both primal simplex and a basic interior point method in MATLAB as part of taking a linear programming class. The main obstacles in primal simplex are making sure that you implement Phase I and Phase II correctly, and also that you implement an anticycling rule correctly. The main obstacles in implementing an interior point method for linear programming tend to be more about implementing the iterative method correctly, and scaling the barrier parameter accordingly.You can find a more complete discussion of the pros and cons of each algorithm in a textbook on linear programming, such as Introduction to Linear Optimization by Bertsimas and Tsitsiklis. (Disclaimer: I learned linear programming from this textbook, and took linear programming at MIT from Bertsimas' wife.) Here are some of the basics:Pros of simplex:Given $n$ decision variables, usually converges in $O(n)$ operations with $O(n)$ pivots.Takes advantage of geometry of problem: visits vertices of feasible set and checks each visited vertex for optimality. (In primal simplex, the reduced cost can be used for this check.)Good for small problems.Cons of simplex:Given $n$ decision variables, you can always find a problem instance where the algorithm requires $O(2^{n})$ operations and pivots to arrive at a solution.Not so great for large problems, because pivoting operations become expensive. Cutting-plane algorithms or delayed column generation algorithms like Dantzig-Wolfe can sometimes compensate for this shortcoming.Pros of interior point methods:Have polynomial time asymptotic complexity of $O(n^{3.5}L^2\log{L}\log\log{L})$, where $L$ is the number of bits of input to the algorithm.Better for large, sparse problems because the linear algebra required for the algorithm is faster.Cons of interior point methods:It's not as intuitively satisfying because you're right, these methods don't visit vertices. They wander through the interior region, converging on a solution when successful.For small problems, simplex will probably be faster. (You can construct pathological cases, like the Klee-Minty cube.)