Strictly speaking, a bathroom backsplash is not necessary. However, that being said, it is generally considered a good idea. Backsplashes prevent both water damage and stains from soap and water on the walls behind them. Not only does it prevent the stains, but it also makes for easier cleaning. Updating your vanity and putting in a backsplash can increase the value of your home, as the biggest rooms that sell are the bathrooms and kitchens.
There are a couple of ideas that sometimes substitute for backsplashes. The first one is using a high-quality glossy paint that resists water. While this will help with both cleaning and water damage, it still pays to have a backsplash. Water can seep behind the sink and sit. Even the most waterproof of paints will have problems with constant contact with water, and damage can still occur.
Another idea that some people use is caulking. This still leaves the upper part of the wall vulnerable to damage and staining from soap and water. Additionally, there can be seepage that gets below and around the caulking. And, caulking would have to be redone on a regular basis as it does wear away.
Tile is also sometimes used instead of a backsplash. This can be a reasonably effective way of protecting the walls from both water damage and staining. Just make sure the seams have been sealed well using a tile sealant. Otherwise, as with caulking, water can seep in between the tiles. Even with the seal, you can still get staining between the tiles.
In short, while not necessary, why would you not use a backsplash. It is simply the most effective way to prevent damage and staining to your walls. This upgrade to your bathroom is well worth the price, and can be easily self-installed.