What are the construction differences between a 1% transformer and a 5% transformer?
From ABB's Transformer Handbook, 3e:


  3.9 Short Circuit Impedance
  
  Users have sometimes particular requirements regarding the
  short-circuit impedance. Such requirements may be determined by:
  
  
  parallel operation with existing units
  limitation of voltage drop
  limitation of short circuit currents
  
  
  The transformer designer can meet the requirements in different ways:
  
  
  The size of the core cross-section. A large cross-section gives a low impedance, and vice versa.
  A tall transformer gives low impedance and vice versa.
  
  
  For each transformer there is, however, a smaller range which gives
  the optimum transformer from an economic point of view, that is the
  lowest sum of the manufacturing costs and the capital value of the
  losses.


The 'short circuit impedance' mentioned above is the transformer's percent impedance. The above quote says that the transformer's impedance can be varied by changing the construction of the core.

Note that a transformers' impedance is mostly inductive "leakage reactance", i.e. magnetic impedance. Therefore, the difference between a 1% transformer and a 5% transformer is mostly to do with the design of the transformer's magnetic core. The 1% transformer would require much more iron core than the 5% transformer, and would be physically larger to match, which explains the higher cost.

From J&amp;P Transformer Book, 12e:


  In Chapter 1 it was explained that the leakage reactance of a
  transformer arises from the fact that all the flux produced by one
  winding does not link the other winding. As would be expected, then,
  the magnitude of this leakage flux is a function of the geometry and
  construction of the transformer....
  
  Since reactance is a result of leakage flux, low reactance must be
  obtained by minimising leakage flux and doing this requires as large a
  core as possible. Conversely, if high reactance can be tolerated, a
  smaller core can be provided.




The conductor resistance (i.e. copper winding resistance) is typically small, 1/10th of the total impedance or less. The guideline given in AS3851 is that power transformers of less than 10 MVA may be considered to have X/R = 10.