What is the use of Google’s “Google Search” Button?

Wonderful observation :) Tried it without Javascript as @J_rgen suggested. Might as well be for the same reason. A fallback for other browsers etc.

Also if you consider from a experience aspect, not having a button would seem somewhat incomplete. Not a perfect reason, but might as well be.

There are three reasons:

1. You have JavaScript disabled on your browser.

2. Your internet is slow and/or has high latency to Google's servers. (Try this! It disables instant search)

3. You use a browser that doesn't support instant.

[Bonus] It has been there always and removing it would confuse people in addition to breaking search on above scenarios.

Because users will get confused

If the search button is removed users will think something is wrong with Google and this will lead to a lot of confusion and abandonment of the search process. This complies with the consistency heuristic.

For those who have disabled instant search

Pradeep is asking the question for the case when the instant search is enabled. However, not every user uses this features so the buttons are left for those who haven't enabled instant search.

Novice users (elderly people) who have no computer experience whatsoever might find it useful.

They might not know that pressing the Enter button will show them the results, so the "Google Search" button stays there as it helps this segment of users.

You will be surprised to see what percentage in the world population haven't sit on a computer and don't know how to use the keyboard.

Firstly, Google Instant is a hidden feature and they provide no cues for it before you begin typing. They can remove the button after the user has begun to type because at this point instant search is exposed to the user. Having no search button available before exposing a hidden feature would be potentially confusing to the user.

Secondly, Google Instant is an optional feature and can be switched off in your search settings. If you have the feature switched off you would need the search button.

Thirdly, as others have already pointed out Google Instant isn't always available. The button needs to be available in case Google Instant is on but doesn't work.

Affordance

Don Norman and James Gibson have differing views on the details of what 'affordance' means but, loosely put, it's allowing the user to perform an action in a way that is obvious to them or making the way to perform that action obvious.

Norman says:

"...the term affordance refers to the perceived and actual properties of the thing, primarily those fundamental properties that determine just how the thing could possibly be used. [...] Affordances provide strong clues to the operations of things. Plates are for pushing. Knobs are for turning. Slots are for inserting things into. Balls are for throwing or bouncing. When affordances are taken advantage of, the user knows what to do just by looking: no picture, label, or instruction needed." (Norman 1988, p.9)

and Gibson says:

"an action possibility available in the environment to an individual, independent of the individual's ability to perceive this possibility" (McGrenere and Ho, 2000).

Microsoft products used to use a triplicate model: most primary actions could be performed by either selecting the action from a menu, clicking a button/icon, or activating a keyboard shortcut. This may be what Google is doing with their search: Those users familiar with using the button can do so and users familiar with hitting enter can do so - neither set of users will be impeded by the interface.

This was tested by Google, as you may imagine.

The thing is that they wanted something that works from an UX point of view while keeping the good ole style we're used to. Basically, you have 2 main sets:

Set 1: Have Google Instant activated

Set 2: Have Google Instant disabled

No matter which set you belong to, you need to see the same page. The user behavior and preferences will create the flow, as in

if $instant--> do something else --> do something else 

you could argue

"ok, I have my settings in place and I chose "Google Instant Enabled", so why is it still showing the buttons?"

and the keyword here is consistency. The addition of those buttons won't add anything to your experience, and it won't cause any harm since they will disappear. But the page is consistent for everybody, so the buttons stand.

Furthermore, the "I'm feeling lucky" button which is part of Google brand still exists and it works, so taking off the other button will leave you with just... I'm feeling lucky, which is a (supposedly) random search. And a very costly feature, as we're at it!. Hence, you leave the button and everything works out great for everybody