What is the difference between "shopping online" and "online shopping"?
Solution 1:
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’