Ques. Am planning to buy a smartphone and my budget is Rs 20k-25k which mobile should i buy am confused between htc desire 820, samsung note 3 neo or second hand samsung galaxy note 3 which also comes in abt 25k...
Ans. At this price point we will only suggest you two phones.
The Nexus 5 and The htc desire 820
The Nexus 5 has the most pure android experience you can get. Samsung's touch wiz UI is prone to slowdowns with time . Even a Samsung galaxy s5 slows down and you need to clear the RAM from time to time.
Another option that you can look for is the ASUS zenfone 5 although bit on the cheaper side but with 2gb of RAM it is almost at par with the big behemoths. Get a cheaper phone and then you can upgrade to another phone next year in the same amount of money that you would otherwise spend on a flagship phone. Lumia 730 is also a fast smooth phone which is the best for music lovers because of its mix radio service.
Lenovo s860 also has good reviews and a very long battery life with its 4000mah battery plus it's a dual sim.
Coming to the choices you have mentioned :
HTC desire 820 has a Octacore 64 bit processor which is future ready for a great andoid L( lollipop) experience
It comes with a 13 mp rear camera + 8 MP front camera for selfies
htc with the snapdragon processor beats Samsung note 3 neo completely, as the samsung comes with a slower exynox processor.