Great question! It really depends on what you mean by "similar". For example, you might say two substances are "similar" if they have similar physical and chemical properties. In that case, it is not generally true that substances made of the same atoms will be similar. For example, the graphite (pencil lead) in your pencil is made of pure carbon. But so is diamond! The reason graphite and diamond are so different even thought they're both made up entirely out of carbon is that their atomic structure is very different. Graphite is made up of sheets of carbon atoms; these sheets can slide easily along one another, which is what allows your pencil to write (it leaves behind a trail of graphite as you rub it on the paper). The carbon atoms in diamond, on the other hand, are all strongly bonded to one another, which is what makes diamond such a hard and tough material.

The point here is that it's not so much the atoms that a substance is made of that determine its physical and chemical properties, but also its molecular structure.

Now, water and hydrogen peroxide happen to have some similar physical properties (they have similar densities, they're both clear liquids at room temperature, etc.). But because they have different molecular structures, they have very different chemical properties: water is a stable molecule, while hydrogen peroxide is unstable due to the bond between the two oxygen atoms (this is why over time, hydrogen peroxide will give off oxygen gas and turn into plain old water). This unstable oxygen bond makes hydrogen peroxide very reactive: it can explode if you heat it enough, and it can cause chemical burns if you get it on your skin. So from a chemical standpoint, water and hydrogen peroxide would not really be considered "similar" compounds.