Depending on what the occasion you are dressing up for, the difference between a great costume and a "so so" outfit always lies in the details and the accessories. But of the different "special effects", there really is nothing that really puts a costume over the top like a great wig or beard or some combination of artificial hair. Not only are they fun to wear, it is the facial hair and wig that makes it harder to tell that you are not the character of your costume and that you are in fact, you. And creating that illusion is what wearing a costume is all out.
You should know that a wig that is made to be used as part of a costume is not the same as the professional hairpieces people buy to replace natural hair in daily life. Wigs for costumes are not made to be worn all the time and they are almost always made of artificial materials, not real hair. But the upside is if you buy your costume, a wig for that costume lasts longer because it is synthetic and you can wear it lots of times and perhaps even adapt it to different costumes.
A powdered wig that would go well with a renaissance costume is an example of a very versatile wig that can be used for lots of costumes. A wig that you might use for a renaissance fair could also be used to participate in a reenactment of an early American historical scene or to become a part of the Victorian court with a costume of a barrister or politician from that time period. If you use that kind of costume a lot such as part of your membership in a renaissance society, it makes sense to buy a good wig for use many times each year.
Usually wigs that are for use with costumes are also not as tough as hair pieces designed to be part of just about any weather condition and to be worn every day all day. So you want to take some care not to expose them to harsh conditions if you want your costume apparel to last for along time and take you to many events. Of course, if you are very particular about the costume because you are a purist about the time frame or just to make the costume as accurate as it can be, wigs to fit the look can be customized for you. But the cost of that kind of quality will be high so look at pre-made wigs first to see if they will give you the look you want.
The fun thing about wigs that you can buy for various costumes is they are not that expensive so you can afford to have several to go with the same costume. The result will be that you might take one costume and give it many different personalities just by changing out the wig. The cost is affordable and having variety just adds to the fun of creating exciting costumes to show off at your next party or period outing.