The main contaminants from a nuclear weapon and a nuclear reactor are the same things: fission products (the "halves" of heavy atoms that remain after you split them), and heavy actinides (un-fissioned uranium and/or plutonium). You can think of these as "residues" of the bomb or reactor. But there were significant differences in the two cases.
The Little Boy bomb dropped on Hiroshima contained 64 kg of fuel, of which ~1 kg fissioned. So that's roughly 63 kg of heavy actinides and 1 kg of fission products. The Fat Man bomb dropped on Nagasaki contained ~6 kg of fuel, of which ~1 kg fissioned. So that's 5 kg of heavy actinides and 1 kg of fission products. The fission products are the fiercely radioactive things immediately after the blast — even though you could fit 1 kg of fission products into your pockets, they're radioactive enough to melt you if they happened to be next to you (about ~14 trillion Curies, for those keeping track at home). When the bombs detonated, their residues were inside the fireball, which rose and became the head of their mushroom cloud. But because they were both detonated high above the ground (to maximize the blast effect), the clouds held the residues for a relatively long time. That gave the most radioactive parts of it time to decay away, and also meant that when the residues did "fall out" of the cloud, they were diffused sparsely, so no single place on Earth got a lot of them. By allowing that decay and diffusion, it meant that the amount of residual radioactivity that anyone could be exposed to was small-enough to not be noticeable, and there was no long-term contamination.
Chernobyl, by contrast, was a nuclear reactor with about 192 tons of uranium fuel inside of it, and toward the end of its fuel load. At least 5% of its fuel and fission products escaped outside during the accident. So that's around 10 tons. Of that, at most 2% of the fuel could be fission products because of its enrichment level. Let's say 1%, just to be safe. So that's maybe 90 kg of fission products that were released (again, compare to the 1 kg of the bombings), plus a lot of uranium-238 (tons) and some heavy actinides. These are just back-of-the-envelope numbers, but it gives you a sense of comparison between the two cases. Aside from the initial explosions that breached the core, the graphite in the fuel was on fire, sending it into the air. But it wasn't as hot as an atomic bomb, and the residues were not fully vaporized as in an atomic bomb. So you've got this not-that-hot, not-that-small stream of radioactive particles coming out of a reactor. They're going to go up, and then very soon come right back down again. So the areas downwind of the reactor got heavy doses of the fuel residues, in high concentrations.
You can still visit Chernobyl today without any ill effects, and small numbers of people could probably live there without detectable influence on outcomes. If you had large numbers of people living there, especially vulnerable populations like pregnant women and children, you'd expect to see the cancer and birth defect rates increase. It's detectably radioactive, but not give-you-radiation-sickness radioactive, at this point. There are occasionally still-very-radioactive artifacts out there, so people tell me; discarded equipment from the cleanup that got unusually high levels of contamination on them. And one should take care not to invite trouble, like eating things that might be contaminated, and getting the remaining residues inside one's body that way.
By contrast Hiroshima and Nagasaki have no detectable radioactivity whatsoever; they're at baseline. If the atomic bombs had been surface or near-surface detonations, would they be radioactive today? A bit — one can see this directly at the Trinity test, which was a Nagasaki-style bomb detonated on a 100 foot tower. You can visit the site today (I was there last summer), and it's detectably more radioactive than baseline. I spent maybe 5 hours there, and by my measurements it seemed like I had picked up about as much radioactivity on the site as I had taking an airplane to New Mexico from the New York City area. So not much. But again if you multiplied that by large numbers of people (esp. vulnerable populations) over large amounts of time, you'd expect to see health problems at a higher level than normal.
To get a comparable contamination to Chernobyl, though, you'd need to release many more fission products no matter what from a weapon. 90 kg of fission products, to use our rough number from above, would correspond to about 1.5 megatons of fissioning. So a 3 megaton hydrogen bomb contains at least that much (the rest would be from fusion reactions, which are not contaminating in the same way).
TLDR: The atomic bombs produced a relatively small number of contaminants, that were kept high in a cloud for long-enough for the worst to decay and dispersed over a relatively large area, diffusing the contamination; the Chernobyl accident released a relatively large number of contaminants, that were immediately contaminating downwind and near the site.
This is more appropriate for /r/askscience... but I'll give you the technical answer here with some interesting history that mitigated some of the horror.
The issue of safety basically comes down to the amount of exposure you will receive per unit time, which itself is a function of the half-life of the kind of fissile material among other considerations. For example exposure is both a function of where it happened, the atmospheric conditions during the disaster, and the kind of radiation from the source including iodine-131, caesium-137, strontium-90, and plutonium-241 -> americium-241. A cursory review of current levels in the Chernobyl Exclusion Zone with now well established sampling areas:
Where Sv is Siervets. 1 Siervet = 100 rems. uSv or one micro Siervet is one millionth of a Siervet
It's worth noting that parts of the exclusion zone are open under supervised guidance including a memorial upwind of the initial disaster.
Another consideration is where the radiation is and where it went. The Chernobyl disaster happened on the ground, but intense fires threw radiation, dust, dirt and other matter (colloquially called fallout) downwind far and wide through the Soviet Union and into Europe. Had the containment floor melted, it would have effectively contaminated water tables on a continent scale. The Soviet Union used hundreds of thousands of "volunteers" to clean up the surrounding area, temporarily cap reactor 4 and with the help of the West, built a more permanent containment coffin with a life-span to effectively control radiation for the next hundred years.
The bombs dropped on Japan were a completely different technology using different fissile material. In the case of Hiroshima, the weapon was a uranium gun-type bomb (detonation to cause sufficient critical mass for fission), code named "Little Boy." The bomb was detonated in the air to cause maximum damage from the pressure wave and electromagnet radiation damage (IR, light, microwave, and gamma-wave) plus the resulting heat. Prevailing winds threw most initial beta-particles and subsequent fallout away from the island. Nagasaki was slightly different. It was not the primary site. Because of a failing fuel pump on the bomb delivery aircraft "Bockscar" a secondary target was chosen... Nagasaki. The bomb dropped, code named "Fat Man" was a plutonium-based (which is technically more deadly and powerful at 21 kilotons), and about 3 km from the intended target of a weapons factory. The explosion was confined to the Urakami Valley protecting much of the city by its geography.
TL;DR: The bottom line is that a series of unrelated events, weather, happenstance, bad luck, and geography made Chernobyl a much different kind of disaster than the atomic bombings of Japan during the end of WWII.
It is also worth noting that "modern" thermo-nuclear hydrogen bombs are yet another technology allowing for damage measured in megatons of TNT. Advancements made in the 60's, 70's and 80's allowed for warheads to be conical in shape to create MRVs (multiple re-entry vehicles) with the idea that some would be decoys to confuse detection. Ideal shapes are spherical. Conical shapes are aerodynamic and can be more compactly fitted into ICBMs
Radioactive contamination comes from radioactive isotopes, most of the dangerous ones for humans in these contexts are fission products. Things like Cesium-137, Iodine-131, Strontium-90. These isotopes are unstable and they decay, as they decay they emit high energy particles or gamma rays, that is the radiation they emit. There are two ways that the presence of these in the environment can be a health hazard to individuals. One is that if you get close enough to a big enough chunk of these materials (which might not be very large at all for some of the more active isotopes) your body will be exposed to radiation from it. Some of this radiation can penetrate into your body and damage molecules in your cells. With high enough doses this radiation can damage the DNA of cells. This can cause cell death but it can also introduce mutations. With very high acute radiation damage there is a risk of the radiation killing enough cells to cause organ and tissue damage, which can become lethal. At lower doses the main risk is DNA damage which can cause enough mutations to result in a higher probability of cancer.
The second way that these can cause physical damage is if they become ingested, absorbed, or inhaled into the body. Some of them (like I-131 and Sr-90) are metabolically active, and can become incorporated into organs or bones. Others simply pass through the body or can become lodged in the lungs. Being within the body gives them a much greater opportunity to damage human tissues and over time they can cause cancers, radiation sickness, etc.
Let's talk about the difference between Chernobyl and the Nagasaki/Hiroshima nuclear weapons. The first major difference is the amount of fission products released. These bombs had much less fissionable material within them than the Chernobyl reactor. The reactor was loaded with many tonnes of material, the bombs only kilos. The amount of fission reactions (and thus the amount of fission products produced) that had occurred in both "fuels" (the bombs and the reactor) was also very different. For the Plutonium "Fat Man" bomb dropped on Nagasaki the amount of energy released was 21 kilotons (TNT equivalent), or 88 terajoules. Chernobyl was a 1000 megawatt electrical powerplant driven by 3200 megawatts of thermal power produced by the reactor. That's 276 terajoules per day of fission energy. Which means that the amount of energy and also the quantity of fission reactions (and radioactive fission products) that the Nagasaki bomb produced (in a microsecond) was equal to what the Chernobyl reactor produced over an operating period of 8 hours.
Because the Chernobyl reactor had been operating for years the amount of fission products (radioactive fallout) that had built up in the reactor was much greater than was produced by the WWII-era nuclear bombs, by an enormous margin. And this was added to by the massive power excursion that caused the accident.
Just as importantly, the ways the bombs and the reactor deposited their radioactive debris into the environment were very different. When the bombs were detonated their entire assembly (the nuclear fuel, the casings, everything) was vaporized in about a microsecond. After which the enormous energy from the fission reactions was deposited into the atmosphere, resulting in a hugely destructive fireball, heat flash, radiation pulse, and blast wave which destroyed their target cities. However, because the debris itself was vaporized, and atomized, it mixed thoroughly with the environment. Some of this material was carried up into the stratosphere by the mushroom clouds and thoroughly dispersed into the upper atmosphere. Even in the case where some of it came back down to earth in the form of "black rain" which caused acute radiation exposure to the people on the ground, because the radioisotopes had been vaporized they were able to be washed away and diluted over time by rain, wind, ocean currents, etc. This was aided by the fact that the bombs were exploded in the atmosphere instead of near the ground. Over long periods (many years) not much of the radioactive fallout from the bombings has remained at either Hiroshima or Nagasaki, which is why they are today places that people live and work instead of exclusion zones.
The situation is completely different with the Chernobyl disaster. The reactor was not vaporized, it was disrupted, and caught on fire. The result was not, primarily, atomized radioisotopes that could be diluted but chunks of material (especially ash) which settled onto the surrounding countryside.
Some of the fallout from Chernobyl was of the sort that could be more easily distributed throughout the environment, which raised the immediate risk to other folks around the world but over time those radiosotopes were diluted and became "just" a slight increase in everyone's background radiation exposure on Earth (as did the Hiroshima and Nagasaki radioisotopes, and those from atmospheric nuclear testing). The real problem was that the ash and solid particles from the reactor in the nearby areas around Chernobyl simply settled to the ground and stuck there. They won't easily wash away or diffuse into the air, they're stuck there for geologic time scales.
If you entered the exclusion zone it wouldn't be like entering the most dangerous portions of the reactor complex today, you wouldn't stumble across an area where the radiation exposure to you would give you either radiation poisoning or even raise your risk of cancer much. And many people do continue to enter the exclusion zone today. The real risk is ingesting or inhaling any of those particles. If you inhale a teeny, tiny dust mote of graphite or fuel rod debris from Chernobyl it can get lodged in your lung for years where it will sit and quietly radiate your lung tissue with high energy particles, raising the chance of developing lung cancer. This is why it's not safe to live and work in the exclusion zone. If people did most would be fine, but a lot more than normal would develop cancer, it's a probability game with bad odds. And there's no easy way to clean up that material other than just scraping off an entire layer of surface material and then putting it somewhere else (which they have done in some areas).