Necessity of static block in Java

Constructor is invoked while creating an instance of the class.

Static block is invoked when a classloader loads this class definition, so that we can initialize static members of this class. We should not be initializing static members from constructor as they are part of class definition not object

I first want to highlight one thing thing from your question:

the constructor does the same thing, do all the necessary stuff when a class is first initialized

This is incorrect. A constructor does all the initialization necessary when an instance of a class is created. No constructors execute when the class itself is first loaded into memory and initialized (unless an instance of the class happens to be created as part of the class initialization). This confusion (between initializing a class and initializing instances of the class) is probably why you are questioning the utility of static blocks.

If a class has static members that require complex initialization, a static block is the tool to use. Suppose you need a static map of some kind (the purpose is irrelevant here). You can declare it in-line like this:

public static final Map<String, String> initials = new HashMap<String, String>();

However, if you want to populate it once, you can't do that with an in-line declaration. For that, you need a static block:

public static final Map<String, String> initials = new HashMap<String, String>();
static {
    initials.put("AEN", "Alfred E. Newman");
    // etc.
}

If you wanted to be even more protective, you can do this:

public static final Map<String, String> initials;
static {
    Map<String, String> map = new HashMap<String, String>()
    map.put("AEN", "Alfred E. Newman");
    // etc.
    initials = Collections.unmodifiableMap(map);
}

Note that you cannot initialize initials in-line as an unmodifiable map because then you couldn't populate it! You also cannot do this in a constructor because simply calling one of the modifying methods (put, etc.) will generate an exception.

To be fair, this is not a complete answer to your question. The static block could still be eliminated by using a private static function:

public static final Map<String, String> initials = makeInitials();

private static Map<String, String> makeInitials() {
    Map<String, String> map = new HashMap<String, String>()
    map.put("AEN", "Alfred E. Newman");
    // etc.
    return Collections.unmodifiableMap(map);
}

Note, though, that this is not replacing a static block with code in a constructor as you proposed! Also, this won't work if you need to initialize several static fields in an interrelated way.

A case where a static block would be awkward to replace would be a "coordinator" class that needs to initialize several other classes exactly once, especially awkward if it involves dependency injection.

public class Coordinator {
    static {
        WorkerClass1.init();
        WorkerClass2.init(WorkerClass1.someInitializedValue);
        // etc.
    }
}

Particularly if you don't want to hard-wire any dependence into WorkerClass2 on WorkerClass1, some sort of coordinator code like this is needed. This kind of stuff most definitely does not belong in a constructor.

Note that there is also something called an instance initializer block. It is an anonymous block of code that is run when each instance is created. (The syntax is just like a static block, but without the static keyword.) It is particularly useful for anonymous classes, because they cannot have named constructors. Here's a real-world example. Since (unfathomably) GZIPOutputStream does not have a constructor or any api call with which you can specify a compression level, and the default compression level is none, you need to subclass GZIPOutputStream to get any compression. You can always write an explicit subclass, but it can be more convenient to write an anonymous class:

OutputStream os = . . .;
OutputStream gzos = new GZIPOutputStream(os) {
    {
        // def is an inherited, protected field that does the actual compression
        def = new Deflator(9, true); // maximum compression, no ZLIB header
    }
};