The advantage of the Spring singleton is that Spring takes care of the logic of making sure you have one (presuming you always go through Spring to get it). The disadvantage is that it isn't a singleton as you can instantiate the object manually. So it relies on conventions. This isn't a problem in practice though. Making the decision amount to whether you are using Spring for other things.
One advantage of having a Spring singleton is that it's much easier to replace by for example a mock implementation for testing than a static Java singleton. You just change your Spring configuration to replace the real one with a mock version.