- dankuwel ("thank you" to an older person, with respect, formal),
- dankjewel ("thank you" to a person your age or similar or younger, formal),
- bedankt ("thank you" either young and old, little less formal, but still formal),
- 'dank je' ("thank you" (literally), which is the most informal of them all, but still okay to say in any situation). Notice that you can also say "dank u", which is "thank you" to an older person, more informal than dankuwel.
"Nee, bedankt." is just a nice and formal way to write things down. In English, when you write 'bedankt' down in a formal way and with nicely used interpunction you get "No, thanks." as well. On a regular day though you might get to see more informal sentences with less interpunction (nee bedankt / no thanks).