Difference between "visa not required" and "visa on arrival"?

Visa on arrival (VOA) means that on arrival, you will need to queue up at a visa counter, apply and pay for a visa, which gets pasted into your passport on the spot, before you go to immigration. (Usually, anyway, since separating the money handlers from the immigration guy reduces the opportunities for graft, but a few countries merge everything to one desk.)

VOA is often limited to certain entry points (eg. international airports, major land borders), so for border crossings where it's not available, you will need to apply for a regular visa in advance.

No visa required means that you can proceed from the plane to directly to immigration, and will not need to pay anything. If no visa is required, you are generally free to enter the country from any legal border crossing and by any means.

To add to jpatokal's answer: You should pay close attention to countries where citizens of some foreign countries do not require a visa, while citizens of other foreign countries require a VOA. In such cases, sometimes the difference is just money, but sometimes it's actually stricter immigration rules for those that require VOA.

For example, a friend and I traveled to Thailand a few years ago. I came in on essentially a one-way ticket, but since I was from a country for which Thailand does not require a visa, I went through a "lightweight" version of the passport/immigration check, and nobody ever questioned me about my plans to leave the country. My friend, however, was from a country for which Thailand requires a VOA, and so he went through a completely different immigration process (physically located in a slightly different part of the airport if I remember correctly), was questioned in detail about his onward travel plans, and was required to show an onward ticket.

Another example of a VOA check: I believe that in Nepal, if you're from a country that requires a VOA, they will check passport stamps to count how long you were already in Nepal in that calendar year, and will only issue a VOA for a duration that will not make you exceed a total of 6 months in the year (including your previous visits). And they will charge different amounts for VOAs of different duration (my data on Nepal is a few years old, things may have changed).