Related parties are those where one has a significant influence over the other, normally (but not always) by shareholding.
"Associated" parties according to the PL are pretty much any two parties where there is an association between the two (familial, business, or any other connection the PL cares to determine) irrespective of the accounting definition.
So yes, a related party by accounting standards would also be "associated" by the PL definition, but most "associated" parties defined by the PL would not be associated as defined by accounting standards.
This isn't new, by the way, it is already in the PL rules. And the current rules aren't really a problem for City as they just require such transactions to be at fair value, which is fairly easy to defend.
What they have just done is tighten up the associated party rules with some new rules that we don't know about yet. Last time, they tried to say such transactions wouldn't be approved unless they were supported by two third party bids as comparators (Liverpool's "what was the losing bid?" question when Etihad sponsorship was first signed). That was voted down last time but it gives City more of a problem, if it is indeed the new rule, because getting three bids for each sponsorship is a pain in the arse and, quite frankly, an onerous burden, which is maybe why City would be thinking about challenging it. And they would win, in my view. The ECJ has recently determined that regulators should keep out of commercial arrangements unless they have a very good reason. I think :)
The current (previous) rules are sufficient protection against inflated "associated" sponsorships, imho, although they too may get thrown out under a legal challenge. At the end of the day, sponsorships should be at fair value, no matter who they are from. Any attempt to restrict a club's ability to sign sponsorships with whoever they want because it's a bit disadvantageous to clubs who aren't so well connected, and we all know this is the issue, is very probably anti-competitive.
At least, that's my take on it all, right or wrong :)