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Both bump into and run into mean to 'meet someone by chance'. (Oxford dictionary definitions)

They are interchangeable.

I think I use run into more often. Bump into sounds a little old fashioned or even "weird" to me.

Merriam-Webster's indicates that they are synonyms for each other and does not provide any meaningful distinction between them in terms of the example sentences that you give. So, I can't give an objective answer.

However, I do believe there is a difference in usage and meaning.

To explain, I will first use use a different scenario along with several different sentence constructions.

Bob started a new job last month that requires he wake up at 7 a.m every morning. This is a change for him, as, for years now, he would wake up at 8 a.m.

One morning, Bob leaves his house at 8:30 a.m.

Bob woke up recently. [He woke up just a short time ago.]

Bob woke up lately. [This is ungrammatical.]

Lately, Bob has been waking up at 8:30 a.m. [His schedule has changed from what it used to be a short while ago.]

Recently, Bob has been waking up at 8:30 a.m. [This is technically correct, but lately would be the preferred word as recently sounds a bit odd here.]

In considering these examples, I would say that recently implies a single event that has occurred a short time ago. Lately, however, implies a change in behaviour or circumstances, as well as referring to something that is more prolonged or recurring.

Returning to your actual examples:

I've been very busy recently. [Your basement flooded last night. It was an unexpected event that you've been dealing with to the exclusion of everything else. You think you have it fixed now.]

I've been very busy lately. [You weren't so busy before your child was born. Now, you always seem to have much less time. Your lifestyle has changed.]

Addendum: Following an offline chat resulting from comments, I thought it might be useful to list the verb tenses I believe are compatible with the two words.

Simple Past—"I ran." Recently only, although perhaps lately when used poetically or archaically.)

Present Perfect Passive—"I have been run off (the road when . . .)." Despite my initial misconception, this is not in the Present Perfect Progressive. Both work best with an extended event, like "I have been sick."

Past Progressive—"I was running (when . . .)." Recently only.

Present Perfect Progressive—"I have been running." (Lately would be preferred to recently, although recently is technically correct.)