I love Merchant Republics to death, but they do manage to be hideously powerful yet wonky.
Basically, you rake in money hand over fist. You can control cities without penalty directly, and you can build trading posts as little addons to counties (your own or otherwise) which produce vast sums of money while boosting income for all cities and trade posts in their consecutive region. The end result is that you have the dosh to build everything, wage war forever, hire mercenaries rather reliably, and so on. I really cannot in good conscience claim they're not stronger than feudal assholes based on this raw power of gold.
Additionally, each merchant house has its own family palace, conveniently located wherever your current capital happens to be. In addition to obvious advantages in being unconquerable, this palace provides a wide range of benefits at efficient prices, since you were so lacking in gold before. This allows you to build up a nicely sized army (and get some freebie prestige/piety gain, plus fertility and other minor stat bonuses) without ever leaving home.
However, then the weirdness starts. For starters, merchant republics are ruled by five families. This means you are shackled to four other AI jackasses at all times, and unless you manipulate everything just right these oafs will frequently be your liege. See, the leader of a merchant republic is "elected" based on age, prestige, and amount of money he's dumped into campaigning, so unless you're constantly ladling prestige-generating positions on your relatively old heirs, sooner or later the AI is going to take over and ruin everything. Or at least not be very good at his job. On the bright side, being led by the oldest guy in the realm makes this a rather temporary problem.
Speaking of which, trading posts. They're awesome. You build them in a special trading post slot in coastal counties, even when owned by somebody else, and they generate revenue. They also produce a "trading zone" that boosts trading post and city income by a percentage based on the value of the zone's holdings; these percentages can get rather large rather quickly, though they start to taper as you climb. Contiguous areas held by the same family add up, meaning having a coherent territory gives much greater benefits than a bunch of piecemeal outposts.
Which is where it all goes wrong, because you're competing with four other braindead jackasses for space, and when you start getting more posts than other people they start thinking about declaring war on you to claim them, stealing them through plots, or if they're your liege just straight up redistributing them to "more deserving" families. Which is far from gamebreaking, but it's infuriating to build up a nice coherent trading area only to have some clown snap it in half because he was too stupid to build his own.
Speaking of jackasses and trading posts, merchant republics have totally different family mechanics than feudal realms. Each adult male of your dynasty at court sucks out a portion of your income, further dampening your obscene wealth generation. Though since they don't often spend money, this means it accumulates in particular branches of your dynasty until you end up playing as them more than it's lost. Also, each adult male adds to your trading post limit, which means in absolute theory they pull their weight.
Perhaps more importantly, merchant republics are locked into agnatic seniority succession, so you have less of a family line plus that scheming asshole uncle who's trying to murder your son, and more a gigantic gaggle of cousins twice removed you'll be playing as any minute now, seeing as you're taking the throne at 67. Alternatively, you can just designate an heir, if you'd prefer to select young competent rulers over replacing your council every five minutes. Me, I like everybody getting a turn.
Finally on family matters: Bride prices and prestige. A badly needed money sink, marrying high-class ladies requires obscene amounts of gold to purchase. Also, family leaders are regarded as counts and republic leaders as dukes, so other rulers will tend to snub you as "a mere count" despite having more cash on hand at this exact moment than their entire dynasty's projected decade of income. You also cannot do matrilinear marriages, which cuts down on your options somewhat.
So I greatly, greatly enjoy them myself and would certainly recommend them over some fancy new unit models, but they've definitely got some odd points, both balance and playstyle wise.