I assume you already studied molecular symmetry and point groups.
So, first of all, among the existing point groups, you can have chiral point groups (point groups C1, Cn, Dn) and achiral point groups (all the others). As a matter of fact, chirality is unchanged when it comes to operations such as E or Cn (that equal an even number of reflections). On the contrary, this is not the case when it comes to operations that equal an odd number of reflections (σ, Sn, inversion), as they can convert a right-handed object into a left-handed object. So an improper rotation axis (Sn) implies that the molecule cannot be chiral.
Both molecules that you show here are chiral, as you correctly pointed out.
2-bromobutan-2-ol belongs to point group C1 as it has no symmetry other than the identity (which is the C1 axis that every molecule has and that corresponds to a rotation by 360° that of course brings the molecule back to the same initial position). Being the C1 axis the only element, C1 molecules are defined asymmetric.
The other two chiral point groups (Cn and Dn) are dissymmetric.
trans-1,2-dimethylcyclobutane has a C2 axis that allows you to rotate the molecule by 180° (which is 360°/2, that's why it corresponds to a 2-fold rotation axis) obtaining a situation that is equivalent to the initial one:
Let's see this with colors, so it's easier:
So, in conclusion, since (1R,2R)-1,2-dimethylcyclobutane belongs to the C2 point group, it is "chiral as well as dissymmetric".
Note: The above is only true in a first order approximation. A more detailed look will reveal that the cyclobutane ring is puckered and the C2 axis no longer exists.