The word gather simply means to bring together or come together:
I gathered all my books and put them into my backpack.
A crowd gathered around the TV to watch the World Cup game.
If something is getting faster or stronger, you can say it gathers momentum or gathers strength:
The charity campaign gathered momentum as it spread on social media. or it spread faster and faster
To collect means you accumulate things and keep them as a hobby:
I collect coins from different countries.
The word collect is also used for getting and keeping things in general:
The government is collecting data from its citizens' phone calls.
The bank is using a questionnaire to collect information from its customers.
We're collecting donations for the animal shelter.
The word gather emphasizes the distant starting point of the things. If your child spreads his toys all around the house, you would gather the toys (from the various places) and put them away in the closet. You could also gather flowers from a field; the flowers are separated from each other in the field.
The word collect emphasizes the close ending point of the things - you are keeping them close to each other, in order to use them in the future.