Compare: - I bumped into him on the street. (chance meeting) - I bumped into him on the street. (light collision) - I bumped into him with my shopping cart. (light collision) - My car bumped into the wall. (light collision) - I ran into him on the street. (chance meeting) - I ran into him with my shopping cart. (strong collision) - My car ran into the wall. (strong collision) - I ran into the store. (literally run, not walk) - I ran into the store. (a very short time) - I ran into traffic. (encounter)

They are both the same, meaning either ' to meet someone by chance ' or 'to collide with or hit against something ' 'Bump into ' is more informal ,sort of colloquial English !

The only difference is that sometimes to run into can mean to encounter, I ran into issues with my computer today. You cant use bump into for this because it literally mans to walk into someone/something.

I would say that there is no difference between the two phrases and can be used interchangeably. As for which one is used more, they are both used frequently and it really depends on the speaker. Some people say "bump into" while others say "run into", and a lot of people use both!