In the Sanger approach, DNA would be isolated from the biopsy and would contain both normal alleles and mutant alleles of genes associated with the development of the tumour. If, for example, PCR amplification was then used to derive a sample of a target template region, this material would end up being sequenced as a mixed population: the derived sequence would be an average of that population of sequences, and rare alleles would be masked.
The key difference in most next generation approaches is that the template DNA is "cloned" physically (e.g. by sequestering individual molecules into droplets, or by binding them to a surface) so that the sequences of these individual molecules can be determined in parallel. This approach would reveal tumour-associated polymorphisms when the sequences of the individual template molecules were compared.
Added later as supplementary information.
I have now looked at the paper cited in the question. This quotation from the Introduction bears out my main point about sequencing a mixture of templates differing at just a few key positions.
Although commonly used in many clinical settings, dideoxynucleotide chain termination (or ‘Sanger’) sequencing of PCR products often lacks sufficient sensitivity for detecting mutant alleles in tumor biopsies, where the failure rate has reached 75% in some cases. Gain-of-function oncogenic mutations are frequently heterozygous events or may represent a single allele of an amplified gene; thus, the signal for mutated residues is typically reduced relative to neighboring bases. Moreover, the ability to detect single base mutations or small insertions or deletions in biopsy material by Sanger sequencing depends heavily on sample purity (for example, the extent of contaminating stromal DNA) and genomic DNA integrity. Furthermore, resistance to kinase inhibitors may correlate with low-frequency second-site mutations. These observations underscore the challenges for accurate mutation detection in cancer specimens.
A massively parallel sequencing-by-synthesis approach, ‘picotiter plate pyrosequencing,’ provides a new alternative to Sanger sequencing. This approach relies on emulsion PCR-based clonal amplification of a DNA library adapted onto micron-sized beads and subsequent pyrosequencing-by-synthesis of each clonally amplified template in a picotiter plate, generating over 200,000 unique clonal sequencing reads per experiment. Sequence variants that represent a fraction of a complex sample can be vastly oversampled, thus enabling statistically meaningful quantification of low-abundance variants.