Thanks for your suggestions and inputs LiHongMSFT and Bruce (SqlWork.com).
I forgot to mention that, even the LOCALHOST was unable to connect to port 1433 in the present scenario. In previous versions of SQL (2014) though it was connecting Remotely but wasn't happening with localhost.
I have finally figured it out. It is the TCP/IP port for "IPAll" portion which was not specified to use 1433 leading to this connectivity issue with localhost as well as from Remote Hosts. Once I specified the port 1433 in IPAll settings and restarted the SQL Server service, both connections worked successfully.
Thanks to All.
Below is the screenshot of that particular setting which made all the trick possible...!