indesign equivalent of ms word's "Don't add space between paragraphs of the same style" feature
The short answer: no.
The useful answer: you can do this by making use of "Next Style" in the Paragraph Style definition:
Set up the Paragraph Style for (as an example) body copy with no space before/after. Call it "Body" (or whatever you like).
Set up a Paragraph Style for your lead paragraph. Lead paras often have slightly different styling from the rest of the body copy, but don't have to. Under "Based On" select your body copy style. Under "Next Style" select "Body," or whatever you named that style.
When you paste text, select all and in the Paragraph Styles panel right-click the "Lead Para" style and choose "Apply Lead Para, then Next Style".
For repetitive formatting of text in individual frames, a killer shortcut is to set one up as above, then make it a new Object Style. In the Object Style definition check "Paragraph Styles" and "Apply Next Style". You can then drop text into a new frame and apply all the formatting with one click on the Object Style.
You've defined that your Body style has no extra space. The only time there's extra space around Body is when it's next to something which isn't Body.
So put the space in the other styles — your Headline, Subhead, Scene, whatever. Any running Body copy will have no spaces. Any Headlines will have them.