A standard example is the following: The push out of $D^1\hookleftarrow S^1 \hookrightarrow D^1$, where the arrows are boundary inclusions, seems to be $S^2$. Now in $hTop_*$ the two 1-discs $D^1$ represent the same object as the point $*$, so the diagram $*\hookleftarrow S^1 \hookrightarrow *$ is the same in $hTop$ and its push out would be the homotopy class of a point, but $S^2\not\sim *$. Homotopy colimits are not categorical, but make use of the model structure and tell you how to build "homotopy-correct" colimits in your model structure. I think this is one of the main reasons why one is happy to have not only the existence of a certain homotopy category, but also the model structure on the localizer. So you can do computations in the model category with the help of the model structure and watch the results in the homotopy category.

A standard example is the following: The push out of $D^2\hookleftarrow S^1 \hookrightarrow D^2$, where the arrows are boundary inclusions, seems to be $S^2$. Now in $hTop_*$ the two 2-discs $D^2$ represent the same object as the point $*$, so the diagram $*\hookleftarrow S^1 \hookrightarrow *$ is the same in $hTop$ and its push out would be the homotopy class of a point, but $S^2\not\sim *$. Homotopy colimits are not categorical, but make use of the model structure and tell you how to build "homotopy-correct" colimits in your model structure. I think this is one of the main reasons why one is happy to have not only the existence of a certain homotopy category, but also the model structure on the localizer. So you can do computations in the model category with the help of the model structure and watch the results in the homotopy category.