Si(OH)4 vs. H4SiO4 vs. SiO2.2H2O
First, my answer here is relevant: What is chemical formula of Bauxite?, and also follow links inside.
Regarding your question specifically, there is not much difference. However, there are some conventions that come into play here.
The three formulae describe compounds that stoichiometrically have four atoms of oxygen and hydrogen for each atom of silicon.
$\ce{SiO2.2H2O}$
This notation simply says that for each component of &quot;silica&quot; there is a single component of &quot;water&quot;. This notation is often used for solids, and never for dissolved species in aqueous solution. In some cases, this can denote that this entire thing has a crystalline structure, in which there are actual molecules of $\ce{H2O}$ held together with &quot;molecules&quot; of $\ce{SiO2}$. This is often called water of crystallisation, but this is not the case here because hydrated silica (which is what this thing is) is not crystalline, but is amorphous.
$\ce{Si(OH)4}$
This is a slightly more simplified way of writing the formula for hydrated silica, withour referring to specific chemical components (such as water or silica). In some cases this indicates that the crystal structure contains hydroxide groups ($\ce{OH-}$), but again, as hydrated silica is amorphous this is not the case here.
$\ce{H4SiO4}$
This is the formula for &quot;silicic acid&quot;, which when in solid form, is exactly the same as &quot;hydrated silica&quot;. However, this notation is often used when discussing hydrated silica in aqueous solution chemistry, as you correctly inferred. This does not necessarily mean that it is an actual acid, at least in the sense we think of when discussing things like sulfuric or phosphoric acid. We have a bottle of commercially acquired &quot;silicic acid&quot; in our lab. It's not an acid - it's just some hydrated silica, but written in the &quot;acid&quot;-type notation.

Something to remember here, is that some of these types of notation imply that the hydrogen occupies a specific structural site in the crystal lattice. The first implies molecular water, the second implies hydroxide groups, the third implies hydrogen protons.
This is misleading, because in fact hydrated silica can contain all three structural types of hydrogen simultaneously, and figuring out exactly which and their relative abundance is often of little importance in most studies. Adding to the confusion is that the compound is not really crystalline (like opal). And finally to confuse you even more, a ratio of 2-to-1 of water to silica isn't required, and it can vary. In fact, sometimes the formula is written as $\ce{SiO2.nH2O}$ because we simply don't know (and don't care) exactly how much water is in there. Finally, to add even more confusion, some groups use one notation while others use another for exactly the same thing. The same author can even use two different notations in two different papers!