This is a nice puzzle--- but the answer is simple: the composite bosons can occupy the same state when the state is spatially delocalized on a scale larger than the scale of the wavefunction of the fermions inside, but they feel a repulsive force which prevents them from being at the same spatial point, so that they cannot sit at the same point at the same time. The potential energy of this force is always greater than the excitation energy of the composite system, so if you force the bosons to sit at the same point, you will excite one of them, so that the composing fermions are no longer in the same state, and the two particles become distinguishable. The scale for this effective repulsion is the decay-length of the wavefunction of the composing fermions, and this repulsion is what leads matter to feel hard.
The reason you haven't heard this is somewhat political--- there are people who say that the exclusion principle is not the cause of the repulsive contact forces in ordinary matter, that this force is electrostatic, and despite this being ridiculously false, nobody wants to get into the mud and argue with them. So people don't explain the fermionic exclusion principle forces properly.
If you have a two-fermion composite which is net bosonic, like a H atom with a proton nucleus and spin-polarized electron, when you bring the H-atoms close, the energy of the electronic ground state is the effective Hamiltonian potential energy for the nuclei. When the nuclei are close enough so that the electronic wavefunctions have appreciable overlap, you get a strong repulsion. You can see that this repulsion is pure Pauli, because if the electrons have opposite spins, you don't get repulsion at short distances, you get attraction, and the result is that you form an H2 molecule of the two H atoms.
You can see this exclusion force emerge in an exactly solvable toy model. Consider a 1d line with two attractive unit delta function pontetials at positions a and -a, each with a fermion attached in the ground state. Each one has an independent ground state wavefunction that has the shape $exp(-|x|)$, but when the two are together at separation 2a, the two states are deformed, and the ground state energy for the fermions goes up. The effect is quadratic in the separation, because the ground state (one fermion) goes down in energy, and the first excited state goes up in energy, and to leading order in perturbations, the two are cancelling when both states are occupied. To next leading order, the effect is positive potential energy, a repulsion. This potential is the effective potnetial of the two delta functions when you make them dynamical instead of fixed.
The maximum value of the repulsive potential in this model is exactly where the model breaks down, which is at a=1. At this point, the ground state is exp(-2x) to the left of -1, constat between the two delta functions, then exp(2x) to the right, with energy -2, and the first excited state is constant to the left of -1, a straight line from -1 to 1, and constant past 1, with energy 0. The result is a net energy of -1 unit. This is half the binding energy of the two separated delta functions, which is -2.
This effect is the exclusion repulsion, and it reconciles the fermionic substructure with the net bosonic behavior of the particle. You can only see the substructure when the wavefunction of the boson is concentrated enough to have appreciable overlap on the scale of the composing fermion wavefunctions, and this is why you need high energies to probe the compositeness of the Higgs (or for that matter, the alpha particle). To get the wavefunctions to sit at the same point to this accuracy, you need to localize them at high energy.
If two atoms differ only by the spin of their nuclei, then their individual properties will be almost identical, but their collective properties will be extremely different.
Chemists often consider individual atoms (or, more often, molecules). In the case of individual hydrogen and deuterium atoms, their electronic properties are identical (except for the hyperfine spitting of their electronic energy levels due to spin-spin coupling between the electrons and the nuclei). Practically speaking, the only important difference is the mass difference due to the extra neutron.
But once you put a bunch of them together and lower the temperature enough that quantum effects (specifically multiple-occupancy of energy levels) become significant, their extremely different many-body properties become manifest. For example, neutral hydrogen atoms are bosons and can in principle Bose-Einstein condense, while neutral deuterium atoms are fermions and will instead form a free Fermi gas (to a decent approximation), which has extremely different properties. That's why it's extremely important that cold-atom experimentalists trying to form Bose-Einstein condensates get the right isotopes of the atoms that they're trying to condense.
Best Answer
Why are fondamental particles either bosons or fermions?
A particle specie is either a boson or a fermion depending on how the wave function changes when permuting two particles of the same specie. A general two particles wave function $\Psi(x_1,x_2)$ can be acted on by the operator $P$ which permutes the two particles. Of course, $P^2\Psi(x_1,x_2) = \Psi(x_1,x_2)$ so $1$ is an eigenvalue of $P^2$ and so the only two possible eigenvalues for $P$ are $\pm1$, $+$ for bosons and $-$ for fermions.
Why are composites also either bosons or fermions?
To answer this, I will cite the excellent answer to the question "How to combine two particles of spin $\frac{1}{2}$?". The main equation to remember is following with it's interpretation: $$ (2j_a+1)\otimes(2j_b+1) = \bigoplus_{i=1}^n(2j_i+1), $$
On the left-handed side, the object describes the hilbert space of two particles, one with spin $j_a$ the other with spin $j_b$. The particles are decoupled so they have a fix total spin and can have any corresponding projection.
On the right-handed side, the object describes the hilbert space of a single composite particle which have can change total spin $j_i \in \{|j_a-j_b|, |j_a-j_b+1|, ..., j_a+j_b\}$.
Both spaces are related by a change of basis. What you need to see is that the total spin, even if it changes, is either an integer or an half-integer. If both particles are fermions or bosons, the composite particle is a boson, but else, the composite particle is a fermion.