Q:

Does the tongue have different taste zones?

I've seen quite a few of these "tongue maps", showing discrete borders between taste zones and often conflicting setups, like these two:

Anecdotally, I can feel all tastes all over the tongue, more or less equally well, so I don't really understand what these map are supposed to indicate. 
Does the tongue have different taste zones as shown?

A:

I remember learning the tongue map in school, but it's a myth.

From The New York Times:

In a study published in the journal Nature in 2006, a team of
  scientists reported that receptors for the basic tastes are found in
  distinct cells, and that these cells are not localized but spread
  throughout the tongue. 
That said, other studies suggest that some
  parts may be more sensitive to certain flavors, and that there may be
  differences in the way men and women detect sour, salty and bitter
  flavors.

Origin of the myth:

The original myth stems back to the early 1900′s when a German
  reseacher named Hanig published data on taste sensitivity of different
  areas of the tongue. 
The differences in sensitivity he reported were
  real — but they were so slight as to be of no practical significance.
  Nobody bothered to check or refute it until many years later, when the
  idea was already firmly rooted in our popular consciousness, and
  textbooks.
Source: Bartoshuk, L. M. 1993. The biological basis of food perception and acceptance. Food Qual. Pref. 4:21-32

More:

Live Science - The Tongue Map: Tasteless Myth Debunked
How Stuff Works - How Taste Works 
University of Washington - That's Tasty
Aroma Dictionary - Challenging the Tongue Taste Map
Stephen Fry - QI: Tongue Map

A:

Linda Bartoshuk’s The Biological Basis of Food Perception and Acceptance (Food Quality and Preference 4, 21–32, 1993)
gives a detailed historical account of the tongue map that is mostly accessible to laypeople. I will only quote the important parts here (boldface mine):

The tongue map with ‘sweet’ on the tip, ‘bitter’ on the back, etc., dates back to the PhD thesis of Hänig […]
    However, it is important to note that Hänig did not plot the actual sensitivity values.
    Instead he plotted rather impressionistic curves meant to show the rate of change from one locus to the next.
    […]
    Saltiness was perceived approximately equally on all loci so Hänig did not include it in his figure.
    […]
Edwin Boring […] discussed Hänig’s thesis […] in 1942.
    Boring did not reproduce Hänig’s summary sketch but rather calculated the actual sensitivities by taking the reciprocals of the average thresholds given in Hänig’s tables.
    Then for each stimulus, Boring divided each of the sensitivity values by the maximum sensitivity achieved for that stimulus […].
    Boring then smoothed the curves a bit and labeled the ordinate ‘sensitivity’ without providing actual values.
    […]
On Boring’s figure, there is no way to tell how meaningful the sizes of the variations are on the ordinate.
    Boring’s graph led other authors to conclude that there was virtually no sensation at the loci where the curves showed a minimum and that there was maximum sensation where the curves showed a maximum and so we have the familiar tongue maps […].
    Even with the ordinate labeled […], the meaning can be confusing. 
    Consider an example from the sweet data.
    At H [the base of the tongue], the sweet sensitivity was 0·3 while at A [the tip of the tongue] it was 1.
    This looks like an impressive difference until we recall what the sensitivity scores mean.
    In fact, they simply mean that at H the threshold was about one-third what it was at A.
    To see what this reflects, look at Fig. 5 [which shows the distribution of threshold across subjects, which spreads across several orders of magnitude].
    […]
    Note that this is a relatively small change.
Collings reexamined the threshold variation in 1974.
    Her results differed from those of Hänig in some regards […]; however, in one very important particular Collings and Hänig agreed.
    There were variations in taste threshold around the perimeter of the tongue but those variations were small.
    Essentially, sweetness, bitterness, saltiness, and sourness can all be perceived on all loci where there are taste receptors.

To summarise, the sensitivity for different tastes (mostly sweet and bitter, by the way) slightly varies around the tongue, but the variation is very small, e.g., when compared to the variation of sensitivity across humans.
Discrete tongue maps are a myth originating from an unlabelled axis and misinterpretation.
As a sidenote, the classical tongue map can be easily debunked by placing a grain of salt on the tip of your tongue.

