What is the real advantage of India's demonetization if it does not solve the problem of black money in India?
What was the logic of introducing a note of higher denomination (₹2000)?
Let's provide a precise definition of the terms in your question for proper context.
Demonetisation refers to the withdrawal of Legal Tender status of the ₹500 and ₹1000 Mahatma Gandhi series of bank notes on November 09, 2016, allowing members of the public to exchange these for the Mahatma Gandhi (New) Series of ₹500 and ₹2000 bank notes which had a different size and design until December 30, 2016 (source: RBI press release that has a broken link to a Finance Ministry Gazette Notification).
This move was announced by Prime Minister Modi in a televised address on November 8 at 2015 hrs IST. Approximately 86% of India's cash economy took the form of these two banknotes. Several rules restricted the ability of the public to freely deposit or exchange old notes.
The Reserve Bank of India's stated objective was that this action was carried out In order to contain the rising incidence of fake notes and black money.
This move was carried out with a great deal of secrecy (though there are some poorly sourced reports of this being untrue).
No official success/failure metrics were given for this "demonetisation", but I try to assess the impact as objectively as I can.
Tackling the stock of fake currency: The new notes introduced security features which are harder to counterfeit. It is unclear how this addresses the original problem since the counterfeits look(ed) similar to genuine notes, even though no security feature ha(d) been copied. However, this move would clearly invalidate the stock of existing counterfeit notes already in circulation.
While depositing the old notes, banks were required to provide adequate number of note counting machines, UV Lamps, note sorting machines etc. at their counters to take care of the work load and timely detection of counterfeit notes, and were required to track and report all depositors (read detailed circular here).
Unfortunately, I have not seen a any data disclosing the measured impact of demonetisation. The RBI is still counting banknotes.
Tackling the stock of black money: The State Bank of India had estimated that about ₹1.5 trillion will not be disclosed by the individuals and this will purely be a extinguished currency liability. Extinguished currency liabilities can then be reintroduced into the money supply. However, by December 30, banks had already received ₹14.97 trillion of the ₹15.4 trillion that was demonetised (97%). Thus, it does appear that there is no easy win here.
Other purported benefits:
Adoption of digital payment schemes: All digital payment schemes had a huge boost in traffic and adoption, with the most prominent example, Paytm claiming 170 million users, with a traffic increase of 435%, and a 250% increase in overall transactions and transaction value.
Reduction in human trafficking and terrorism: Though there was some anecdotal evidence pointing to a decline, this was not substantiated by data from the National Crime Records Branch
Negative effects: I try to stick to the headline numbers here, since you asked about the advantages. This move was a monetary shock, which led to reduced discretionary spending and disruption in production activity, in the short term. In the medium-to-long-term, the impact depends on the ability of firms to absorb the shock. In the Growth Outlook for 2017-18, the RBI expected the impact of the liquidity shock to largely dissipate by mid-February, and with rapid remonetisation, pent up demand was likely to boost consumption demand in 2017-18.
The Reserve Bank of India's Sixth Bi-monthly Monetary Policy Statement, 2016-17 on Feb 08, 2017, estimates a 0.33% reduction in GDP growth for the year 2016-2017 can be attributed to this demonetisation.
In the unorganized sector, about 80% of micro, small and medium enterprises were apparently negatively impacted, [based on a statement by Anil Bhardwaj, the secretary general of the Federation of Indian Micro and Small and Medium Enterprises].
A private think tank, CMIE estimates that 1.5 million jobs were lost for various reasons, mostly due to side effects of demonetisation.
Short-term political impact: The BJP has won landslide victories in the most of the elections after November 8, 2016. A recent poll indicates 73% approval ratings for the central government. As it fulfilled a stated election promise, and signaled seriousness of intent, the decision is still popular. Occasionally, the popularity has been attributed to schadenfreude.
See the RBI's preliminary report on the Macroeconomic Impact of Demonetisation or a more digestible article in the online press for a more detailed coverage.
To answer your other question, ₹2000 is not a very high value (though it is currently the highest denomination Indian bank note, it is worth only about USD 60 in PPP terms). The ₹1000 note was reintroduced in the year 2000. Since then the INR has weakened 75% in PPP terms, which makes the 2017 ₹2000 note a fairly reasonable denomination for the times. Before that, the ₹500 banknote denomination was introduced in October 1987, so a new highest denomination note was overdue.
Being a higher value banknote, it sped up the remonetisation, effectively requiring fewer banknotes to be printed. It is said to be only a short-term bridge that will be phased out soon. It isn't clear if all of this is being made up by spokespersons as they go along; I haven't seen any written evidence / minutes-of-meetings detailing any of these decisions.
What is the real advantage of Demonetization if it does not solve the problem of black money in India?
There was no advantage of Demonetization.
Apparently, Modi wanted to do something against corruption, but, also wanted to keep his voters, party members, and donors unharmed as corruption comes mostly from politicians and businessmen. In the end, nothing improved, and only sufferers were poor people who had to stand in the queue for hours to get their currencies changed from the bank.
Another reason may have been that Modi wanted to show some kind of feat or charisma by popping a new idea. But, that proved to be a futile attempt in the end.
A better approach would have been to attempt to restrict the use of cash and to increase use of cards. But, that should have been introduced gradually and by stages. In that way, the government would have been able to track the paths of flows of money very easily.
What was the logic of introducing a note of higher denomination(Rs 2000)?
Currencies of higher denominations is generally a sign of inflation.
In the case of India, the introduction of a higher currency note isn't clear to me.