What is the difference between "shopping online" and "online shopping"?

Both sentences have the same meaning and they are grammatically correct because 'online' is both an adjective and adverb (https://en.oxforddictionaries.com/).

In the 'shopping online' there's an adverb, in the 'online shopping' there's an adjective.

According to Oxford Living Dictionary:

Adverb

 : By means of the Internet or other computer network.

‘shoppers would rather pick up the phone than do business online'

Adjective

: (of an activity or service) available on or performed using the Internet or other computer network.

‘online banking’