'Run into' means 'meet by chance', and suggests more of a shock than 'come across'. We can use it when we meet people by chance, e.g. 'I ran into an old school friend in town yesterday. We hadn't seen each other for years'. We can also use it for unexpected difficulties 'The project ran into a few problems'. 'Come across' means 'find', and is used mainly for things and information. 'I came across some old photos when I was looking through my grandmother's attic' or 'I came across some interesting facts when I was browsing the internet'. 'Come across' is gentler than 'run into'.

I would use 'ran into' when referring to a person - i ran into an old friend yesterday. However, I would say I came across some old letters. However, I would also say I ran into some problems- I am not sure that is particularly helpful. I can't think of a simple rule that will help you. Hopefully someone else has a better explanation.