Why does splitting a nucleus release energy but breaking molecular bonds requires energy? The title pretty much says it all. Why is it that energy is required to break molecular bonds but energy is released when the bonds within the atomic nuclei are broken.
 A: I think that you may be comparing two distinct concepts.  For both exothermic and endothermic reactions (or related but not the same, exergonic and endergonic reactions), breaking a bond still requires at least a small amount of energy to overcome the activation energy barriers.  This is the energy barrier that forms the boundary of the stable minimum energy structure.  Now whether the next stable configuration (energy minimum) outside of that barrier is higher or lower energy depends on the direction of the reaction.
For example, in biological systems the molecule adenosine triphosphate (ATP), is a common energy currency for the cell, acting somewhat like a rechargeable battery.  (Many different proteins have a common ATP binding site, much like different electronic devices have a common battery compartment.)  ATP releases its energy through a breaking of molecular bonds to form adenosine diphosphate (ADP) and a free phosphate.  The released chemical potential is used to drive another chemical reaction that otherwise wouldn't occur.
In this "discharging" of ATP, the net change of internal energy is negative because energy is released, somewhat like your fission example.  However, ATP when dissolved in water is stable, meaning that it is energetically perched in a local energy minimum, well above the final energy state of ADP.  There is an activation energy barrier preventing its spontaneous hydrolysis.  Additional activation energy is needed to break the bond, but this energy is almost immediately recovered in moving to the other side of the barrier.
Importantly, and to the point of your question, activation energy is also required to reform the ATP molecule.  The reverse reaction to recharge the battery by reforming ATP from ADP and free phosphate, also has an energy barrier.  In this case, the activation energy barrier is even greater than the difference in energies of ATP and ADP, as the energetic perch of ATP is below the peak of the activation energy barrier.
It appears that a similar concept exists for fission reactions, called the fission barrier. See:
https://en.wikipedia.org/wiki/Fission_barrier
A: First, both binding energies are the energy required to separate the constituent (atoms or nucleons) of the subject (molecule or nuclei).
Quoting Wikipedia,
Bond energy and bond-dissociation energy are measures of the binding energy between the atoms in a chemical bond. It is the energy required to disassemble a molecule into its constituent atoms. This energy appears as chemical energy, such as that released in chemical explosions, the burning of chemical fuel, and biological processes.
Nuclear binding energy is the energy required to disassemble a nucleus into the free, unbound neutrons and protons it is composed of. It is the energy equivalent of the mass defect, the difference between the mass number of a nucleus and its measured mass.
$$B({}^A_ZX)=[Nm_n+ZM({}^1H)-M({}^A_ZX)]c^2$$
This would be much clear with an example, Suppose we scatter gamma rays (photons) from deuterium gas and look for the breakup of a deuteron into a neutron and a proton
$$\gamma +d\rightarrow n+p$$
Then if we suppose $K_n=K_p=0$ (violating the momentum conservation but it's alright)
$$hf_{min}=m_nc^2+M({}^1H)c^2-M({}^2H)c^2=B_d$$
So you have to give energy (that is through photon) to break the nuclei and this energy will appear as additional rest energy.
A: Well, there is the example of Acetylene (systematic name: Ethyne)
When Acetylene is under a sufficiently high pressure then a sufficiently hard shock can trigger detonation. That is, while there is a high threshold to dissociation of the Acetylene molecule, the energy that is released in the subsequent reaction is larger than that threshold. The reaction becomes a runaway reaction: explosion.
(It is no coincidence that an acetylene torch is the hottest burning torch. You get a double whammy; the acetylene is burned with oxygen, together with dissociation of acetylene being exothermic.)

Comparison with nuclear fission.
The fissionable isotopes of Uranium and Plutonium do have a threshold energy to dissociation. The higher that threshold, the longer the half life of that isotope.
A nucleus is inherently stable if the amount of energy that is released upon fission is smaller than the threshold energy necessary to bring the nucleus to a state from which it might proceed to fission. Such a nucleus can spontaneously go to such a close-to-fission state from time to time (random fluctuation) but it will always fall back to the base state.
Fissionable isotopes can proceed to fission, because the energy released exceeds the threshold energy.

(Incidentally, in the case of explosion of a fission device the runaway is executed differently. Capture of a neutron changes the long half life isotope to a highly unstable isotope. In the case of the Uranium and Plutonium isotopes used: the fission of that unstable isotope also results in emission of several neutrons. That is how the runaway is achieved.)
