Can I use both abilities of Tainted Field? What's the difference?
They can both be used but only one can be used at a time. This is because to use either ability, you need to tap the card and it can not be tapped twice

The second ability lets you produce colored mana where the first ability only lets you produce colorless mana.

Note also that you must control a swamp in order to use the second ability.

So when you play that land, you can immidiatly tap it for colorless mana if you do not control a swamp. If you do control a swamp, then you can tap it for either a white or a black mana or a colorless mana. If you do not need specifically a white or a black mana, then yes both abilities produce one mana which can be used to pay a colorless cost.

This is of course important because unless you are playing with primarily artifacts, you are going to need mana of a specific color.

This card imparticular allows you to play one land that can produce a single mana of two different colors under specific circumstances. This is very usefull in a deck that has cards of the two colors in it.



Comprehensive rules:

All lands are permanents with no color that have at least one mana ability.

Basic lands have the ability to produce when tapped a single mana of their type. 


  305.6. The basic land types are Plains, Island, Swamp, Mountain, and Forest. If an object uses the words "basic land type," it's referring
  to one of these subtypes. A land with a basic land type has the
  intrinsic ability "{T}: Add [mana symbol] to your mana pool," even if
  the text box doesn't actually contain that text or the object has no
  text box. For Plains, [mana symbol] is {W}; for Islands, {U}; for
  Swamps, {B}; for Mountains, {R}; and for Forests, {G}. See rule
  107.4a. Also see rule 605, "Mana Abilities."