I want to compare diferences between homotopy types of simplicial complexes, $\Delta$-complexes and CW-complexes. (to avoid confusions I'm using the definitions, notations und constructions of simplicial complexes and $\Delta$-complexes from Greg Friedman's An elementary illustrated Introduction to simplicial sets
In this excellent answer Eric Wolfsey explaines the fundamental differences between the construction of simplicial complexes, $\Delta$-complexes and CW-complexes. The key point is the presence of different levels of restrictivity of glueing maps of the boundary of the simplices.
Therefore we have proper inclusions of classes
$$ \text{simplicial complexes } \subset \Delta \text{-complexes} \subset \text{CW-complexes} $$
Here I would like to discuss how far these classes differ from each other as on level of homotopy types. Let $A$ some interesting class of topological spaces, eg simplicial complexes or CGHS (compactly generated Hausdorff spaces). I introduce the non standard notation
$$\text{Ho(certain class of topol. spaces A)}:= \{X \text{ top. space } \ \vert \ \text{ there exist a space } Y \in A \text{ such that } X \simeq Y \} $$
The natural question is, if there is something known about inclusion relations between $ \text{Ho(simplicial complexes) }, \text{Ho(} \Delta \text{-complexes)} $ and $\text{Ho(CW complexes) }$?
As Eric Wolfsey pointed out in his answer, the inclusion $ \text{simplicial complexes } \subset \Delta \text{-complexes} $ is proper, since there exist $\Delta$-complexes which are not constructable as simplicial complexes and simular of the CW-complexes, since $\Delta$-complexes allow tremendously more freedom degrees in glueing procedure. By same reason, the CW-complexes form a bigger class that the $\Delta$-complexes.
But what is known about eg the inclusion $ \text{Ho(simplicial complexes) } \subset \text{Ho(} \Delta \text{-complexes)} $ between the sets of homotopy types?