I am a self admitted Civ noob, but I think what makes civ better or worse in comparison to others are two things.
1. How situational the UA is. Look at Denmark - their Viking Fury is great... if you can get just right placement on map with just right amount of coastal cities you can abuse. Otherwise whole game can play out without you ever being able to get any real use out of it. Or Polynesia - on archipelago, continents or terra ability to sail across ocean right away is pretty sweet. On pangea? You won't get much mileage out of that one... Then there are UAs, that will only support a single VC. Bushido will find some use on virtually any map, even if only to deal with barbarians more efficently. But will it support your attempt to flood opponents with manga and anime till they become completly subservient to you (i.e. cultural victory as Japan)? Or will it make for faster Farnsworth moment (science victory)? Nope. On the other hand Solidarity will always be useful, as long as you don't fail your Science 101, by supplying whichever SP you need for your current goals - be it export of enough vodka to make opponents drown themselves (why do you think you get any tourists as Poland?
) or riding them down with those Winged Hussars or joining Mr. Twardowski*. Similarly Babylon's and Korea's science bonuses can be used to get a tech supporting VC of your choice faster, thus making both those civs unconditionally viable.
Long story short, whenever civs UA forces you to play certain way or requires certain map conditions, for you to get any mileage out of it, it lowers the general strehght of the civ.
2. Unique units/Unique Buildings. Again, those can be more or less useful compared to their standard counterparts or maybe even less useful due to what they replace. Let us again look at example of Japan - fighters see pretty light use in the game, thus making Zeros moderately attractive at best due to how situational they are. Or Norwegian Ski Infantry, whose bonuses are mostly pertinent to tiles that no one will fight for or in, due to their lack of worth. Compare this with China, who gets a maintenance free libraries, that make money as well as science AND the Cho Ko Nu, that are one of strongest ranged units of their period and become only more ridiculous as you upgrade them to Gatlings, MGs and finally bazookas.
I think that restrictivness of UA is more of an issue than UU/UB though, since even unimpressive UAs and UBs at the very least do the job of their standard counterpart and as such can't really hinder you.
*A character in Polish legends, that as a result of deal with the devil ended on the Moon way before Neil Armstrong.