Powering an Arduino with more than 5 V
First of all you should read the tech specs for Arduino Uno here:
https://www.arduino.cc/en/Main/ArduinoBoardUno

You will find this important information about power supply:

Operating Voltage   5V
Input Voltage (recommended) 7-12V
Input Voltage (limit)   6-20V


From documentation also:

Vin. The input voltage to the Arduino/Genuino board when it's using an external power source (as opposed to 5 volts from the USB connection or other regulated power source). You can supply voltage through this pin, or, if supplying voltage via the power jack, access it through this pin.

5V pin.This pin outputs a regulated 5V from the regulator on the board. The board can be supplied with power either from the DC power jack (7 - 12V), the USB connector (5V), or the VIN pin of the board (7-12V). Supplying voltage via the 5V or 3.3V pins bypasses the regulator, and can damage your board. We don't advise it.

So answering you question: yes you should have supplied 12v on Vin. Not on 5v.


  Why does an adapter with 12V input not make the same blown up for arduino like in the previous case?


Because you are pluging 12v in the correct input, the 7-12v jack.

Another important link to read:
http://playground.arduino.cc/Learning/WhatAdapter