A big benefit is that you can make decisions, specifically structural ones, that might be best for your product, but not for the parent. For example, StackOverflow's most likely revenue stream was clearly going to be ads, which was not only a different revenue stream than Fog Creek selling shrinkwrapped products, but could have put Fog Creek into a position where it had a conflict of interest (see Google and Microsoft for excellent examples of that at scale). Spinning it off avoided that. Likewise, it was obvious that Trello was going to need a lot of VC to grow fast, which (at least at the time) ran directly opposite Fog Creek's "bootstrap everything, grow organically, never sell" attitude, so spinning that off also made sense.

As a counterexample, while I obviously don't know that Kiln would have thrived if it had gone a different direction, I do know that being locked into FogBugz' business model, instead of having the freedom to innovate our own, did ultimately stunt our growth quite a bit. FogBugz charged by the user, and so Kiln had to charge by the user, but this really hurt us. First, our costs scaled by repositories' size and count, not by users (which were basically free for us as such), so our costs didn't necessarily have anything to do with our revenue. Second, charging by the user is a horrible idea for an SCM. FogBugz could reasonably provide a clear separation between tools using its data and people using its data (since the only UI was the website), but in the world of DVCSes, it's really tough to provide the same distinction. Many people happily just used things like TortoiseHg and treated Kiln as a code backup service, so having no-website accounts wouldn't have accomplished much. We ended up forced into an awkward position of creating deliberately hobbled modes of access for tools that would by definition be prohibitively obtuse for humans, or telling people to shell out an extra $15/mo just so Jenkins could talk to the thing. Kiln switching to its own business model while remaining at Fog Creek would've been theoretically possible, but when business decisions that might be good for your new product will hurt your legacy product, that's a really tough argument to make. Having Trello and StackOverflow at their own companies completely avoided that problem.

The downside, of course, is redundancies and frayed vision. In a world where Trello and StackExchange remained at Fog Creek, I can imagine Fog Creek being the productivity company, with all of these tools tightly integrated à la Microsoft Office and presenting a coherent vision of how to develop software. You won't get that if you're spread across multiple companies. And, of course, you can end up in situations where, specifically because of all the reasons I pointed out above, two of your companies are going at each other a bit (e.g. Trello vs. FogBugz), which, even if subtle (those products don't honestly actually compete much), means you're spending at least some money competing with yourself. And, of course, you lose out on being able to easily move employees from one company to another, reusing technology amongst multiple companies, etc.

I'm sure there's more, but those are ones I remember seeing while I was still there.