What's the difference between a pentaquark and a proton? The proton isn't just 3 quarks bound by gluons: From here, a proton also has an uncountable lot of sea quarks, antiquarks and gluons and that

The standard shorthand, “the proton is made from two up quarks and one down quark”, is really a statement that the proton has two more up quarks than up antiquarks, and one more down quark than down antiquarks. 

If this is so, then what is the difference between a proton and a pentaquark such as Pc+ which has a quark structure of uudc
 $\bar{\mathbb{c}}$ ? The c and  $\bar{\mathbb{c}}$ could just annihilate, just like the sea which constantly form pairs of sea quarks and sea antiquarks and then annihilate each other to form back to gluons. The proton also could have a c $\bar{\mathbb{c}}$ pair formed in its sea, making them no different from the Pc+. The pentaquark also has two up quarks more than down antiquarks and one more down quark than antidown quarks, just like the proton! Or is there a difference in the extra valence quark and antiquark in the pentaquark and the quarks and antiquarks in the sea?
 A: One is talking of new resonances within  baryonic decay products, hard to see in the usual scattering experiments:


The mass of J/ψ–proton (J/ψ p) combinations from Λb → J/ψpK-decays. The data are shown as red diamonds. The predicted contributions from the Pc(4380)+ and Pc(4450)+ states are indicated in the purple and black distributions, respectively. Inset: the mass of J/ψ p combinations for a restricted range of the K-p mass, where the contribution of the wider Pc(4380)+ state is more pronounced.

What do this data tell us:
The  Λb has as three  valence quarks udb and a mass of 5.620GeV . BUT it displays in its decay modes resonances attributed to penta quarks.
If one could have had a J/Ψ beam of mesons the way one can have a K- beam, and could scatter J/Ψ on protons these resonances would appear in the scattering experiments. As the J/Ψ has such a short lifetime and as a neutral is hard to manipulate in a beam, this is the only way these can be seen and classified.
In the proton itself, there does not exist enough energy for the virtual quark-antiquark pairs of the sea to acquire  a type of "valence" state.
The  sea quarks and gluons in the proton graph in prof. Matt Strassler's site you quote, have different energy contents.

The contribution of sea quarks in scattering experiments is very small as seen by the antiquark distributions. If a valence pentaquark existed within the proton, it would have appeared in the deep inelastic experiments which probe the parton distributions. 
Think of nuclear models where models with alpha particles internal to the nucleus are used. This cannot happen in the proton because of the large energy scales introduced by the strong interactions. 
It seems that we have to accept  the existence of pentaquarks as the statistical significance is high, and cross checks are successful. Pentaquarks are allowed by various theories, but their mass values  have to be determined by experiment, and it seems it has happened.

What's the difference between a pentaquark and a proton?

One has to take the "why" further than "a proton is a bag of quarks antiquarks and gluons" , into "why this unique mass and not a continuum?". The baryon decouplet shows that there are more baryon resonances, so there is not a unique mass, but masses obeying a symmetry, incorporated into the SU(3)xSU(2)xU(1) symmetries of the standard model. This is due to the flavor blindness of the strong interaction, which does not distinguish the flavor quantum numbers.
In the same way that the energy levels of hydrogen do not cover the whole energy spectrum but are discreet, the hadronic masses are discreet, the proton and neutron being the ground state of the baryonic set. There is also the  meson set with  symmetries displayed by their masses. The pentaquark hypothesis follows this logic, that the weak interaction blindness of the strong interactions may allow strong bound states more complicated than three valence ones, and it seems that the hypothesis is valid.
(As an indication of calculational progress , Ab initio calculation of the neutron-proton mass difference)
For example, here is a paper with the possible representations of penta quarks, in analogy with the baryon octets and decuplets:

We study the SU(3) group structure of pentaquark baryons which are made of four quarks and one antiquark.  The pentaquark baryons form 1,8,10,10,27, and 35 multiplets in SU(3) quark model. 

