Acid Catalyzed Hydration of an Alkene Synthesis Problem
You propose to use but-1-ene rather than but-2-ene as a starting material to synthesise butan-2-ol via addition of water to the double bond. There is no reason why it would not happen the way you drew.

However, using but-2-ene allows only one possible reaction pathway. With the relatively high symmetry of the molecule ($C_\mathrm{2h}$, assuming trans), attacking either side of the double bond will lead to an identical product: The molecule’s intrinsic $C_2$ axis will transform one into the other. Thus, it is only possible to generate butan-2-ol.

Starting from but-1-ene, you have two non-equivalent first steps, generating either the primary or the secondary carbocation. These obviously cannot be transformed into each other by symmetry (the alkene’s symmetry is now $C_\mathrm{s}$). Thus, you will get a side product: butan-1-ol. These products will be difficult to separate. If anything, careful distillation would do the job.

As RobChem already noted in the comments, (organic) chemists typically attempt to avoid side reactions and different products, hence why the book suggests using the more symmetric alkene.