Shipment Calculations According to Source and Destination

How we can add this functionality like if we want to add same rate by calculating weight if source and destination is same, but charge different shipment rate if source and destination is different? e.g. if shipment will took place in London and source of product is also London than we have a less but different rate but if source is London but destination is not London than it considers as "Out of City" shipment and charge different rate.

You can use a Shipping Director plugin, there is some options available at marketplace, take a look at this one.

You need to use Warehouse for 'source'.  Out of the box, the shipping provider "Fixed or By Weight and By Total" (Shipping.FixedByWeightByTotal) can handle warehouses.  Set up Warehouses with source address, select a warehouse for each product, and in Shipping Settings, select "Use warehouse location".

If you need more complex shipping scenarios/calculations, Shipping Director also has a "Use warehouse location"; be sure to unselect the nopCommerce one if using the Shipping Director one.