There's been a definite increase in regulations governing public companies in the past few years. Counsel for a public company needs to be at least familiar with the rules even if they don't specialize in corporate work. And they're not alone; the responsibilities for compliance fall on legal, finance, and executive roles. Many companies are even developing compliance departments with the sole purpose to marshal it all and keep everyone on point. Which gives a little breathing room to the non-corporate in house counsel, but doesn't completely remove the added complexity of working for a public company.
There's also an increasing perception that practicing in a private company relieves an attorney of all burdens of knowing and following the required regulations for public companies; and consequently, completely unprepared to work for a public company. I've seen it most predominately when a company is searching for a new (or first) GC. Since I've never met a founder or a VC that doesn't dream of the huge IPO, even private companies want someone with public company experience for fear that one can't gain the necessary experience in a private company.
Of course most founders and VCs dreams of an IPO or big exit, so if you're working for a funded company you have to at least think about putting the controls in place that would ease that process. Additionally, in a funded company the investors will require many of the same type of controls be in place so they feel comfortable giving more money. Moreover, the diligence process for selling a company in lieu of an IPO is often more in depth and a lot more time consuming than the diligence process for conducting an IPO. So an attorney can gain valuable experience even working at a private company.
That said, not all founders have big exit dreams. Some dream of growing a company to a respectable size and serving their customer base for generations. They want nothing more than to live a good life and leave a legacy to their families. Those closely held companies don't usually take outside funding. And they don't really care about following pointless process that don't add real value in at the moment. They're often not interested in acquiring any other businesses and care more about the day to day transactions than corporate transactions. A lawyer coming out of this environment won't be prepared to work for a public company without a good mentor to show them the way.
So all this goes to say, whether working in a public vs private company makes a material difference in either the day to day or your chances of getting that promotion often depend on the companies involved, the leadership requirements and whether or not you've got a good mentor.