what is the difference between `public class` and just `class`?

I have noticed that if don't write public before a class its works same as like a public class.

No it doesn't. Unless it's public, the class won't be visible to other code which isn't in the same package. The default accessibility (which can't be specified explicitly) is that a class (or other member) is only visible to other code within the same package.

You should read the Java Language Specification section 6.6 and the Java Tutorial (Controlling Access to Members of a Class) for more details.

public, protected and private are access modifiers. Public means that the subject may be accessed by any class, protected by subclass, private by the class itself, no modifier means "package protected", so the subject may be accessed by classes from the same package.

Subject is class, method, member variable.