  We present a strict separation between the class of "mismatch free"
self-assembly systems and general aTAM systems. Mismatch free systems are those
systems in which concurrently grown parts must always agree with each other.
  Tile self-assembly is a model of the formation of crystal growth, in which a
large number of particles concurrently and selectively stick to each other,
forming complex shapes and structures. It is useful in nanotechnologies, and
more generally in the understanding of these processes, ubiquitous in natural
systems.
  The other property of the local assembly process known to change the power of
the model is cooperation between two tiles to attach another. We show that
disagreement (mismatches) and cooperation are incomparable: neither can be used
to simulate the other one.
  The fact that mismatches are a hard property is especially surprising, since
no known, explicit construction of a computational device in tile assembly uses
mismatches, except for the recent construction of an intrinsically universal
tileset, i.e. a tileset capable of simulating any other tileset up to
rescaling. This work shows how to use intrinsic universality in a systematic
way to highlight the essence of different features of tile assembly.
  Moreover, even the most recent experimental realizations do not use
competition, which, in view of our results, suggests that a large part of the
natural phenomena behind DNA self-assembly remains to be understood
experimentally.
