There is no difference between the objects; you have a HashMap<String, Object> in both cases. There is a difference in the interface you have to the object. In the first case, the interface is HashMap<String, Object>, whereas in the second it's Map<String, Object>. But the underlying object is the same.

The advantage to using Map<String, Object> is that you can change the underlying object to be a different kind of map without breaking your contract with any code that's using it. If you declare it as HashMap<String, Object>, you have to change your contract if you want to change the underlying implementation.

Example: Let's say I write this class:

class Foo {
    private HashMap<String, Object> things;
    private HashMap<String, Object> moreThings;

    protected HashMap<String, Object> getThings() {
        return this.things;
    }

    protected HashMap<String, Object> getMoreThings() {
        return this.moreThings;
    }

    public Foo() {
        this.things = new HashMap<String, Object>();
        this.moreThings = new HashMap<String, Object>();
    }

    // ...more...
}

The class has a couple of internal maps of string->object which it shares (via accessor methods) with subclasses. Let's say I write it with HashMaps to start with because I think that's the appropriate structure to use when writing the class.

Later, Mary writes code subclassing it. She has something she needs to do with both things and moreThings, so naturally she puts that in a common method, and she uses the same type I used on getThings/getMoreThings when defining her method:

class SpecialFoo extends Foo {
    private void doSomething(HashMap<String, Object> t) {
        // ...
    }

    public void whatever() {
        this.doSomething(this.getThings());
        this.doSomething(this.getMoreThings());
    }

    // ...more...
}

Later, I decide that actually, it's better if I use TreeMap instead of HashMap in Foo. I update Foo, changing HashMap to TreeMap. Now, SpecialFoo doesn't compile anymore, because I've broken the contract: Foo used to say it provided HashMaps, but now it's providing TreeMaps instead. So we have to fix SpecialFoo now (and this kind of thing can ripple through a codebase).

Unless I had a really good reason for sharing that my implementation was using a HashMap (and that does happen), what I should have done was declare getThings and getMoreThings as just returning Map<String, Object> without being any more specific than that. In fact, barring a good reason to do something else, even within Foo I should probably declare things and moreThings as Map, not HashMap/TreeMap:

class Foo {
    private Map<String, Object> things;             // <== Changed
    private Map<String, Object> moreThings;         // <== Changed

    protected Map<String, Object> getThings() {     // <== Changed
        return this.things;
    }

    protected Map<String, Object> getMoreThings() { // <== Changed
        return this.moreThings;
    }

    public Foo() {
        this.things = new HashMap<String, Object>();
        this.moreThings = new HashMap<String, Object>();
    }

    // ...more...
}

Note how I'm now using Map<String, Object> everywhere I can, only being specific when I create the actual objects.

If I had done that, then Mary would have done this:

class SpecialFoo extends Foo {
    private void doSomething(Map<String, Object> t) { // <== Changed
        // ...
    }

    public void whatever() {
        this.doSomething(this.getThings());
        this.doSomething(this.getMoreThings());
    }
}

...and changing Foo wouldn't have made SpecialFoo stop compiling.

Interfaces (and base classes) let us reveal only as much as is necessary, keeping our flexibility under the covers to make changes as appropriate. In general, we want to have our references be as basic as possible. If we don't need to know it's a HashMap, just call it a Map.

This isn't a blind rule, but in general, coding to the most general interface is going to be less brittle than coding to something more specific. If I'd remembered that, I wouldn't have created a Foo that set Mary up for failure with SpecialFoo. If Mary had remembered that, then even though I messed up Foo, she would have declared her private method with Map instead of HashMap and my changing Foo's contract wouldn't have impacted her code.

Sometimes you can't do that, sometimes you have to be specific. But unless you have a reason to be, err toward the least-specific interface.

For Java 9 and beyond, you can make use of Map#of, as well as Map#ofEntries with Map#entry:

Map<String, String> example = Map.ofEntries(
    Map.entry("1", "one"),
    Map.entry("2", "two"),
    Map.entry("3", "three"),
    //...
);

For shorter literals (10 or less entries), you can use Map#of directly (albeit with a slightly confusing syntax of Key1, Value1, Key2, Value2, ...):

Map<String, String> example = Map.of("1", "one", "2", "two");

Prior to Java 9, this would typically only be needed for initializing constant Map objects, for which a static initializer block is also useful:

private static final Map<String, String> MY_CONSTANT;

static {
    Map<String, String> setup = new HashMap<>();
    setup.put("1", "one");
    setup.put("2", "two");
    setup.put("3", "three");
    MY_CONSTANT = Collections.unmodifiableMap(setup);
}

This will allow you to create a constant Map while allowing you to modify the map before it is set.

As pointed out by @Pshemo, there are numerous reasons to avoid double-brace initialization, however the notable exception to the "rule" is typically in constants much like the above example (as numerous anonymous classes would not be made). However, it is still possible to work around the need for double-brace initialization even in this case, so it is overall best to avoid if possible. The only case I can think of this not being possible is within the context of a constant within an enum class, where the double-brace init may truly be the only true way to make this work.