The key difference between glycogen and amylopectin (the main constituent of starch) is not the number of α l,6-glycosidic branches, but their arrangement.
In glycogen branches are successively subdivided, producing a relatively small globular structure that is unable to grow further. It is soluble in an aqueous environment and, with its numerous exposed ends, can be metabolized rapidly — appropriate for animal cells in which energy reserves must be mobilized in response to immediate demands, e.g. for muscle contraction.
In amylopectin there is a long central polysaccharide chain from which branches of limited size extend at intervals. This produces much larger semi-crystalline particles (starch grains), a form especially suited to long-term bulk storage in seeds and tubers.
This is the common feature of glycogen and the amylopectin portion of starch. (The amylose portion is unbranched.) In glycogen there is approx. one branch point per 10 glucose units, whereas in amylopectin the figure is 1 per 24–30 (source: Wikipedia).
The contrasting branching topography of the two polysaccharides, mentioned above, is illustrated diagrammatically below:
This is a two-dimensional representation. In three dimensions the glycogen spreads out in all directions from a central point — actually the primer enzyme, glycogenin. In three-dimensions the amylopectin strands mainly lay side by side.
The illustration below, modified from Bell et al., shows the different shapes and sizes of the macromolecular structures. It should be mentioned that semi-crystalline nature of amylopectin is aided by the helical conformation of the chains.
Rather than providing a précis of the review of Bell et al. (Journal of Experimental Botany, Vol. 62, pp. 1775–1801, 2011) I shall quote from them directly (omitting their citations).
As regards glycogen they write:
Each chain, with the exception of the outer unbranched chains,
supports two branches. This branching pattern allows for spherical
growth of the particle generating tiers (a tier corresponds to the
spherical space separating two consecutive branches from all chains
located at similar distance from the center of the particle). This
type of growth leads to an increase in the density of chains in each
tier leading to a progressively more crowded structure towards the
Mathematical modelling predicts a maximal value for the particle size
above which further growth is impossible as there would not be
sufficient space for interaction of the chains with the catalytic
sites of glycogen metabolism enzymes. This generates a particle
consisting of 12 tiers corresponding to a 42 nm maximal diameter
including 55,000 glucose residues. 36% of this total number rests in
the outer (unbranched) shell and is thus readily accessible to
glycogen catabolism without debranching. In
vivo, glycogen particles are thus present in the form of these limit
size granules (macroglycogen) and also smaller granules representing
intermediate states of glycogen biosynthesis and degradation
(proglycogen). Glycogen particles are entirely hydrosoluble and,
therefore, define a state where the glucose is rendered less active
osmotically yet readily accessible to rapid mobilization through the
enzymes of glycogen catabolism as if it were in the soluble phase.
Regarding amylopectin they write:
Amylopectin defines one of, if not the largest, biological polymer
known and contains from 105–106 glucose residues. There is no
theoretical upper limit to the size reached by individual amylopectin
molecules. This is not due to the slightly lesser degree of overall
branching of the molecule when compared to glycogen. Rather it is due
to the way the branches distribute within the structure. The branches
are concentrated in sections of the amylopectin molecule leading to
clusters of chains that allow for indefinite growth of the polysaccharide.
Another major feature of the amylopectin cluster structure
consists of the dense packing of chains generated at the root of the
clusters where the density of branches locally reaches or exceeds that
of glycogen. This dense packing of branches generates tightly packed
glucan chains that are close enough to align and form parallel double
helical structures. The helices within a single cluster and
neighbouring clusters align and form sections of crystalline
structures separated by sections of amorphous material (containing the
branches) thereby generating the semi-crystalline nature of
amylopectin and of the ensuing starch granule. Indeed the crystallized
chains become insoluble and typically collapse into a macrogranular
solid. This osmotically inert starch granule allows for the storage of
unlimited amounts of glucose that become metabolically unavailable.
Indeed the enzymes of starch synthesis and mobilization are unable to
interact directly with the solid structure with the noticeable
exception of granule-bound starch synthase the sole enzyme required
for amylose synthesis.
The paucity of information on plant starch metabolism would seem to reflect a combination of their being less is known about plant biochemistry, and less general interest because of a general focus on medical and animal biochemistry. Although an animal biochemist myself (and, thus, previously ignorant of the information in this answer) I feel that it is time to redress this imbalance.