What's the difference between "lately" and "recently"?
I've been very busy lately.
I've been very busy recently.
Any subtle differences in meaning?
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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.)