Can a nuclear power plant explode like an atom bomb? Absolutely not. But let's explore.
Atom Bomb Physics
If you take a lump of uranium-235 (the fissionable isotope), under the right conditions this lump will undergo a chain-reaction in a fraction of a second and release energy. A lot of energy.
Luckily, the lump of uranium has to be very large to get this explosion to happen. Here's why: the principle behind the chain reaction is that the fission of a uranium atom by shooting a neutron into it produces several more neutrons, which can fission other atoms. If the lump of uranium is small, too many neutrons are lost to the outside to keep the reaction going. (I'll explore the geometry of fission in another article).
The original research into the possibility of making a weapon indicated that about a ton of material would be needed. In fact, it is believed that the German bomb program during World War II faltered on this point: 2000 pounds of uranium-235 could not be produced before the war would end. The American program eventually found a way to focus neutrons back into the lump of uranium so that a bomb only needed a hundred pounds or so of U-235. ("Uranium Enrichment: How to Make an Atomic Bomb," cosmosmagazine.com).
The products of fission in an atom bomb include a large number of elements in many isotopes. As these products break down by radioactive emissions, they do a substantial part of the damage from the bomb. And because some of the radioactive fission products have long half-lives (they last a long time in the environment), their effects can be felt many years after the initial explosion.
Nuclear Power Plant Physics
A nuclear power plant runs on the same principle as a bomb, but the fission process is controlled to allow an acceptably slow release of energy which can be turned into heat and then electricity. The uranium cannot explode like a bomb. First of all, the uranium is purified much less in a power plant than in a bomb; U-235 makes up only about 5% of the uranium in a power plant (naturally occurring uranium has less than 1% U-235). A bomb requires 90% purification. In addition, the amounts of uranium in power production are not sufficient for a bomb without the sophisticated focusing mechanism.
Radioactive Products of Fission
The fission products of a nuclear power plant are exactly the same as those of a nuclear explosion. These fission products, most of them dangerously radioactive, are considered 'waste;' they are hard to deal with. In the best case, they are cooled (the radioactive process creates a lot of heat), and eventually buried. The 'waste' from a nuclear explosion can never be captured, and remains in the environment until it decays.
In the long run, the primary harmful products in nuclear waste are strontium-90 and cesium-137; these are produced in considerable quantities and have a half-life of about 30 years. That means these radioactive elements remain in the environment for decades. Strontium-90 is particularly dangerous because it seeks bone and can cause bone cancer ("Strontium," epa.gov).
The emission from the recent nuclear accident in Japan is small compared to the fallout from a bomb, but since many authorities believe there is no safe level of radiation ("All Levels of Radiation Confirmed to Cause Cancer," nirs.org), any emission is too much.