Let's take the simplest example: you have an application and you just use the default classloader. You have a class which, for whatever reason, you decide that it should not have more than one instance in the application. (Think of a scenario where several people work on pieces of the application).
If you are not using the Spring framework, the Singleton pattern ensures that there will not be more than one instance of a class in your application. That is because you cannot instantiate instances of the class by doing 'new' because the constructor is private. The only way to get an instance of the class is to call some static method of the class (usually called 'getInstance') which always returns the same instance.
Saying that you are using the Spring framework in your application, just means that in addition to the regular ways of obtaining an instance of the class (new or static methods that return an instance of the class), you can also ask Spring to get you an instance of that class and Spring will ensure that whenever you ask it for an instance of that class it will always return the same instance, even if you didn't write the class using the Singleton pattern. In other words, even if the class has a public constructor, if you always ask Spring for an instance of that class, Spring will only call that constructor once during the life of your application.
Normally if you are using Spring, you should only use Spring to create instances, and you can have a public constructor for the class. But if your constructor is not private you are not really preventing anyone from creating new instances of the class directly, by bypassing Spring.
If you truly want a single instance of the class, even if you use Spring in your application and define the class in Spring to be a singleton, the only way to ensure that is also implement the class using the Singleton pattern. That ensures that there will be a single instance, whether people use Spring to get an instance or bypass Spring.