Constant is the concept, the property of the variable.

final is the java keyword to declare a constant variable.

As other people pointed out, from a semantic/linguistic point of view the expression constant variable is an oxymoron and, as such, we could argue about its correctness.

Quoting the specification, anyway, we can read

A variable of primitive type [...], that is final and initialized with a compile-time constant expression (§15.28), is called a constant variable.

I suppose, hence, that we can accept (and consider correct) this binomial for our purpose.

In java, making something final means that it can't be reasigned to another reference to another instance, but if it's a reference to a mutable class, the mutable values inside the class can still be modified.

For example, a final String is a constant because Strings are immutable in Java, but a final ArrayList means that you cannot assign it to another ArrayList, but you can still add and remove elements to that ArrayList

In C++ marking a member function const means it may be called on const instances. Java does not have an equivalent to this. E.g.:

class Foo {
public:
   void bar();
   void foo() const;
};

void test(const Foo& i) {
   i.foo(); //fine
   i.bar(); //error
}

Values can be assigned, once, later in Java only e.g.:

public class Foo {
   void bar() {
     final int a;
     a = 10;
   }
}

is legal in Java, but not C++ whereas:

public class Foo {
   void bar() {
     final int a;
     a = 10;
     a = 11; // Not legal, even in Java: a has already been assigned a value.
   }
}

In both Java and C++ member variables may be final/const respectively. These need to be given a value by the time an instance of the class is finished being constructed.

In Java they must be set before the constructor has finished, this can be achieved in one of two ways:

public class Foo {
   private final int a;
   private final int b = 11;
   public Foo() {
      a = 10;
   }
}

In C++ you will need to use initialisation lists to give const members a value:

class Foo {
   const int a;
public:
   Foo() : a(10) {
      // Assignment here with = would not be legal
   }
};

In Java final can be used to mark things as non-overridable. C++ (pre-C++11) does not do this. E.g.:

public class Bar {
   public final void foo() {
   }
}

public class Error extends Bar {
   // Error in java, can't override
   public void foo() {
   }
}

But in C++:

class Bar {
public:
   virtual void foo() const {
   }
};

class Error: public Bar {
public:
   // Fine in C++
   virtual void foo() const {
   }
};

this is fine, because the semantics of marking a member function const are different. (You could also overload by only having the const on one of the member functions. (Note also that C++11 allows member functions to be marked final, see the C++11 update section)

C++11 update:

C++11 does in fact allow you to mark both classes and member functions as final, with identical semantics to the same feature in Java, for example in Java:

public class Bar {
   public final void foo() {
   }
}

public class Error extends Bar {
   // Error in java, can't override
   public void foo() {
   }
}

Can now be exactly written in C++11 as:

class Bar {
public:
  virtual void foo() final;
};

class Error : public Bar {
public:
  virtual void foo() final;
};

I had to compile this example with a pre-release of G++ 4.7. Note that this does not replace const in this case, but rather augments it, providing the Java-like behaviour that wasn't seen with the closest equivalent C++ keyword. So if you wanted a member function to be both final and const you would do:

class Bar {
public:
  virtual void foo() const final;
};

(The order of const and final here is required).

Previously there wasn't a direct equivalent of const member functions although making functions non-virtual would be a potential option albeit without causing an error at compile time.

Likewise the Java:

public final class Bar {
}

public class Error extends Bar {
}

becomes in C++11:

class Bar final {
};

class Error : public Bar {
};

(Previously private constructors was probably the closest you could get to this in C++)

Interestingly, in order to maintain backwards compatibility with pre-C++11 code final isn't a keyword in the usual way. (Take the trivial, legal C++98 example struct final; to see why making it a keyword would break code)