I learned the equation
$$\ce{H2O + H+ -> H3O+}$$
And I know $\ce{H+}$ and $\ce{H3O+}$ really mean the same thing. But I am confused as to when I should use $\ce{H+}$ and when I should use $\ce{H3O+}$? Is any one of them a more accurate representation?
In addition to entropid's answer, let's remember why we invoke the hydronium ion $\ce{H3O+}$ in the first place.
We use $\ce{H3O+}$ as a shorthand for $\ce{H+(aq)}$, which looks more like protonated water clusters of the generic formula $\ce{H+.(H2O)_{$n$}}\equiv \ce{H_{$2n+1$}O_{$n$}+}$.
Almost ten years ago, a very interesting paper appeared in Science that examined the structure of these clusters ($n=2,3,4,5,6,7,8$). The linked article has been made freely available by Science, but you need to register for a free account at sciencemag.org.
One of the key points of the article was the determination of the inner structure of these clusters. For $n=1$, you have $\ce{H3O+}$, the hydronium ion, which we are reasonably familiar with. When $n=2$, you have $\ce{H5O2+}$, which has a different structure: the proton is evenly shared between two water molecules: $\ce{[H2O\bond{...}H\bond{...}OH2]+}$. Higher order clusters have these two structures, termed the "Eigen" and "Zundel" ions, respectively, at their cores.
The study used a combination of theory (MP2/aug-cc-pDVZ) and experiment (photodissociation vibrational spectroscopy). They found the for $n=2,6,7,8$, the clusters have a Zundel core, while the Eigen core exists for $n=3,4,5$. Whoa! And it gets more complicated as you increase the number of water molecules in the clusters (might be behind a pay wall) - though it turns out the Eigen ion is slightly more common in larger clusters.
In aqueous solutions, we can have an extensive dynamic network of clusters always in flux. In which case, we have a mix of solvated Eigen $\ce{H3O+}$ ions representing protons closely associated with water molecules and Zundel $\ce{H5O2+}$ ions representing protons in transit.
So, we use $\ce{H3O+}$ to keep things simple, and also to satisfy the needs of Brønsted-Lowry acid-base theory, in which every acid-base reaction needs a proton-donor and a proton-acceptor. Thus, the "dissociation" of acids in water is not really a dissociation, but an ionization reaction - an acid-base reaction in which water is the base and hydronium is the conjugate acid.
$$\ce{HA + H2O <=> A- + H3O+}$$
Of course, it is only appropriate to use $\ce{H3O+}$ in aqueous solution. In other solvents, the "proton" has a different structure. In general, the proton is attached to a basic site on the solvent molecule. For example, in methanol:
$$\ce{HCl + CH3OH <=> Cl- + CH3OH2+}$$
There is no real difference between the two notations. As you know, $\ce{H+}$ cannot exist on its own in solution as it gets hydrated to $\ce{H3O+}$, but in most cases it can be used when writing equations or working with reactions. To my knowledge, $\ce{H3O+}$ is slightly preferred when dealing with acid/base reactions, e.g.
$$\ce{H2O + HCl -> H3O+ + Cl-}$$
while $\ce{H+}$ is used more – for simplicity – with redox reactions, e.g.
$$\ce{Zn + 2H+ -> Zn^{2+} + H2}$$
