This is something that understandably confuses a lot of newly engaged couples. Both help you to make the wedding in your head come to fruition, but each in a different way. To use a simple analogy, imagine that getting ready for your wedding is like building a house. Your Planner is your general contractor, while your Stylist is your interior designer. The planner oversees big picture logistics and all the “sub-contractors” who are involved in making your big day happen. She (or he) helps you find vendors who are a good fit for you, assists you with contracts, creates a timeline and flow for your wedding day, and manages logistics (among many other things). On your wedding day, she will direct your family and wedding party during the ceremony, coordinate all vendors, keep things on schedule, etc.
Your stylist, on the other hand, will focus more on the aesthetics of your wedding. She (or he) will help you figure out what sort of “look” you want your wedding to have, give you ideas, and assist you in things like choosing a color scheme, designing your tabletops, acquiring or designing decor, and ensuring that all the details like stationery, flowers, china, linens, lighting, etc. work well together visually. The day of the wedding, she may be responsible for implementing decor and working with the florist to ensure that everything comes together as planned.
These are by no means a comprehensive lists of what each person does and of course every situation is different. Many times the planner and stylist are one in the same. This is totally fine, but you will definitely want to discuss this up front so that all parties involved have the same expectations. Also, do not assume that if someone is good at one role that they will necessarily be good at the other. There are great wedding planners who may not be particularly creative or share your sense of style, just as there are brilliant stylists who may not know how to negotiate a contract with the caterer or how to make sure your granny in a wheelchair navigates the venue comfortably and successfully. Some more successful stylists will not even take a job unless there is a planner involved.
While I always recommend using a planner regardless of your budget (and even require it at my own venue), a separate wedding stylist may be an expenses that you choose to forgo if design is not a priority for you. Some brides consider styling the most important part of their wedding, while others would rather save their money to use towards catering or a great band. The important thing is to think (along with your fiance) about what aspects of your wedding day are most important before you commit to spending any money, and then make sure that you understand what services are included when you interview planners or stylists. As in all aspects of weddings, clear and open communication will eliminate confusion on down the line.