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I need to replace my Z012 freewheel due to wear on my 12 speed road bike. I can't seem to find a replacement that matches it. The large gear has 28 teeth and the small has 13.
I bought a MF-TZ20 to replace it but the teeth on it are 1/8 in shorter and about 1/3 narrower? The small gear has 14 teeth rather than the 13 on the original. Will this work and if so will it effect the type of chain it will take? I need to change the chain as well.
Really any 6 speed freewheel will work. You'll want to get the tooth numbers close, but they don't have to be exact (your 14-28 will be a fine replacement.) Chain sizing for 6-7-8 speed drivetrains hasn't been significantly changed since they became popular, though the profiles and faces of the freewheel cog teeth have, in most brands, been altered to improve shifting quality (see the image below.) There is no need to worry about the tooth profile being incompatible with a modern 6-7-8 speed chain. 
You will, however, want to replace the chain. This is because as a chain ages and wears, it becomes longer. The chain rollers/bushings/pins are spaced at an exact 1/2", and as it lengthens, they wear the chainrings and freewheel cogs into a hook profile. If you use your old chain on your new freewheel, you'll likely find that it skips when pedaling (hopefully your chainrings aren't similarly worn, but if they are you will know what to replace next.)
Assuming you have an indexed shifter, more important that the width of the gears is the spacing - if that's the same (I expect it will be) then it will fit on the hub and the derailleur will move the correct amount to effect gear changes. If the shifter is not indexed, then it won't be so critical.
You should however replace the chain, as if the cluster is worn out, the chain is likely to be, and the old worn chain will quickly wear out a new cluster. You should carefully look at the chain rings (on the pedal cranks), and replace those (if worn) as well for the same reason.
The tooth height and profile will not make a difference.
The 14 vs 13 smallest gear will mean you high gear is reduced - potentially reducing your top speed, otherwise it not be a problem.
I suggest some light reading over Here
There is one on eBay now, no? I would think many Shimano 6-speed freewheels would work, and there are many on eBay. But there were different spacings. The MF-Z012 was late 1980's, so your rear dropouts should be 126mm or thereabouts, meaning you could go with a 6- or 7-speed freewheel.
