First, due to its greater sensitivity and functional resolution, the functional localization method will a) allow you to potentially discover regions that would be missed with traditional group analyses; and b) see clearer functional profiles, with better differentiation among conditions, for each region you are examining. This is true regardless of whether you are using an explicit "localizer" task, or whether you are applying the GSS-style analysis methods to a dataset without a localizer task (read here about how to do that).
Note also that these advantages apply to traditional fMRI designs that rely on the mean BOLD signal differences across conditions, as well as to designs that use neural suppression paradigms, multivariate pattern analysis methods, or other techniques.
Second, increases in functional resolution are especially critical if you are asking questions about functional specialization (i.e., whether two mental processes rely on the same neural mechanisms). The figure below (from Fedorenko & Kanwisher, 2011) illustrates this point. We extracted the response from two group-level ROIs that are commonly used in the literature (corresponding to Brodmann Areas 44 and 45, i.e., ~ "Broca’s area"), and from subject-specific fROIs for our LIFG parcel, which is similar in its location to these Brodmann areas. As can be clearly seen, both of the group ROIs show weak effects for both the language task (sentences > nonwords), and for a spatial working memory task (hard spatial WM > easy spatial WM). Critically, however, if we focus on language-responsive, i.e., sentences > nonwords, voxels (using a portion of the data) and examine their response to the two language conditions (in a left-out portion of the data) and to the spatial WM conditions, we see a striking degree of specificity such that the language-responsive fROIs show no response to spatial WM (besides, the sentences > nonwords effect increases in size several-fold). Using group-level ROIs in this case would lead us to fundamentally different (and wrong) conclusions. (See this paper for evidence that Broca's area contains both language-selective and highly domain-general subregions.)
And third, perhaps the greatest advantage of functional localizers is that they allow for easier accumulation of knowledge across studies and across labs (if similar localizers are used). For example, imagine you are studying some mental process (e.g., syntactic processing), and another lab is also studying this mental process, with both labs using the traditional group-based methods. Let’s say you conduct a study and discover that some region X is engaged in syntactic processing. And let’s say that the other lab does a study and does not find region X for their syntactic task but instead finds region Y in some other part of the brain, or maybe it finds a region nearby to where you found your region X, but not quite in the same location. Where do we go from here?
In such situations (that are all too common in the field of language research) it is hard to determine which set of results is (more) valid and should be trusted, and/or to decide whether two nearby activations reflect the activity of the "same" region or of two nearby functionally distinct regions. If, on the other hand, both labs can agree on some functional signature that a brain region should demonstrate if we are to think that the region is important for syntactic processing (e.g., a stronger response to sentences than sequences of unconnected words), then both labs can define the relevant region(s) in the same way, thus establishing that they are studying the "same" region(s), and test different hypotheses about these regions. For example, one lab might want to know whether regions that support syntactic processing in language also care about structure in music, and another lab might want to know whether regions that support syntactic processing are sensitive to the frequency of syntactic constructions. In that way, the two labs can relate their sets of findings to each other in a straightforward way. Some would say, "Well, can't we just use coordinates in the common stereotaxic space to relate different sets of findings to each other?" We don’t think this is a good way. For example, as we discuss in our 2009 paper, in a review published in 1998 Aguirre & Farah have argued for a distributed representation of different visual categories (faces, objects and words) in the vental visual cortex based on observing no clear spatial clustering for a set of activation peaks from previous studies investigating visual processing. However, later work has established that in each individual subject different regions in the ventral visual cortex are highly specialized for processing different categories of visual stimuli, suggesting that sets of coordinates in stereotaxic space are not well-suited for asking questions about the architecture of human cognition.