The symptoms of Down syndrome occur due to overexpression of genes present on the duplicated chromosome.
If possessing an extra chromosome meant an equivalent change in gene expression, one would expect to observe 50% more protein production for having 3 rather than 2 copies of a chromosome. However, due to complexities in regulation at the level of transcription, translation, and protein degradation, the actual expression levels vary substantially. These complexities can arise from alleles present on chromosome 21 or elsewhere on the genome. I will present some examples from three categories of variation that are known to contribute to the spectrum of symptoms in Down syndrome.
1. Complexity of chromosome duplication
Down syndrome is sometimes caused not by a complete, but rather a partial trisomy 21 (Antonarakis et al., 2004). These variations can also partly explain the severity of symptoms because some individuals do not possess a full duplication. You mention this in your question, but as you note, this is a fairly rare occurrence relative to full trisomy, so let us consider other contributions...
2. Variation of expression levels in normal individuals and those with trisomy 21
It turns out that among the genes expressed on chromosome 21, mRNA levels vary between normal individuals by as much as 40-fold! (Deutch, et al. 2005; Stranger, et al. 2005) This expression variability can explain the susceptibility of different individuals to trisomy 21, depending on the expression levels of the alleles they possess.
Among the different genes present on chromosome 21, some expression levels are consistently elevated in Down syndrome across individuals, some have overlapping but significantly different distributions (suggesting some Down syndrome patients have expression levels in the normal range and others do not), and others are indistinguishable between patients and controls (Prandini et al., 2007).
Presumably, genes in the first category contribute most to the shared phenotype of Down syndrome, and genes in the second category contribute most to the variation. Perhaps alleles that produce mRNA transcripts at the low end of normal for those genes are less susceptible to the effects of chromosome duplication.
A case study: Amyloid precursor protein
One protein of interest in particular is the amyloid precursor protein, APP, which is also associated with Alzheimer's disease (which shares some phenotypic characteristics with Down syndrome). APP expression varies widely among tissue types and individuals. Therefore, although APP mRNA levels are significantly elevated in Down syndrome individuals, the distributions between controls and Down syndrome are very overlapping; for example, see Figure 2B from the Antonarakis 2016 review.
3. Interactions with genes on other chromosomes
The third contributor to the variation of symptoms is the interaction of duplicated chromosome 21 genes with alleles located on other chromosomes. Just for an example where some of the genetic basis is understood, Down syndrome individuals are susceptible to certain leukemias, which are also associated with specific alleles on other chromosomes (Antonarakis, 2016). It seems that trisomy 21 affects histone modification in the areas of those alleles (Lane et al., 2014) and promotes proliferation of B-cells. Therefore, Down syndrome interacts with those other oncogenes to produce a greater combined risk. Individual with Down syndrome but not possessing the other alleles are less susceptible to the increased risk of leukemia
Similar interactions are likely with other systems that are influenced by Down syndrome, though the full molecular basis of all of those interactions are not fully understood. The Down Syndrome Genomes Project aims to, among other things, discover these other alleles outside of chromosome 21 that contribute to Down syndrome symptoms, which may also help understanding of the contribution of those alleles to other disorders (Antonarakis, 2016).
Antonarakis, S. E. (2016). Down syndrome and the complexity of genome dosage imbalance. Nature Reviews Genetics.
Antonarakis, S. E., Lyle, R., Dermitzakis, E. T., Reymond, A., & Deutsch, S. (2004). Chromosome 21 and down syndrome: from genomics to pathophysiology. Nature reviews genetics, 5(10), 725-738.
Deutsch, S., Lyle, R., Dermitzakis, E. T., Attar, H., Subrahmanyan, L., Gehrig, C., ... & Antonarakis, S. E. (2005). Gene expression variation and expression quantitative trait mapping of human chromosome 21 genes. Human molecular genetics, 14(23), 3741-3749.
Lane, A. A., Chapuy, B., Lin, C. Y., Tivey, T., Li, H., Townsend, E. C., ... & Yoda, A. (2014). Triplication of a 21q22 region contributes to B cell transformation through HMGN1 overexpression and loss of histone H3 Lys27 trimethylation. Nature genetics, 46(6), 618-623.
Prandini, P., Deutsch, S., Lyle, R., Gagnebin, M., Vivier, C. D., Delorenzi, M., ... & Baldo, C. (2007). Natural gene-expression variation in Down syndrome modulates the outcome of gene-dosage imbalance. The American Journal of Human Genetics, 81(2), 252-263.
Stranger, B. E., Forrest, M. S., Clark, A. G., Minichiello, M. J., Deutsch, S., Lyle, R., ... & Deloukas, P. (2005). Genome-wide associations of gene expression variation in humans. PLoS Genet, 1(6), e78.
(note: the two references I have linked here: Antonarakis 2016 and Prandini et al 2007, are, respectively, a nearly direct answer to the posed question that establishes the current state of knowledge about symptom variability, and an original research paper that provides much more detailed genetic analysis of the variability of expression of many relevant genes than would be appropriate for an answer here; I highly recommend them both for further reading on the subject)