Note that I am no virologist but am explaining to the best of my knowledge
and for practical use from a variety of reports and my own understanding from my background in Nursing and Health Informatics. I think it is worth answering because the reasoning comes up a lot with skeptical friends and family who might question if it's even "worth" getting the vaccine, or "worth" maintaining measures to prevent infection spread.
To answer your questions in a nutshell--the increased viral load of all variants can be infectious in both vaccinated and unvaccinated people (that is to say, it's not that 1 is infectious and 1 is not--any can be potentially infectious).
My main takeaway from the article you shared is that flip-flopping on policies regarding masks may contribute to further spread, since vaccinated folks can still contract/spread the Delta variant. While vaccinated folks are more likely to experience the more mild or asymptomatic symptoms of Delta COVID-19, the inconsistency of mask-enforcement makes infection spread pernicious.
To further go into my answer for your questions--it is possible that what makes Delta a variant of concern is that it has mutated to replicate much faster than previous variants we've observed. A faster replication rate of SARS-CoV-2 in our cells contributes to a higher viral load in a shorter amount of time, giving the virus an exponentially greater opportunity to be shed.
With higher viral loads, it’s believed that people who contract SARS-CoV-2 shed more virus, making it easier to transmit the virus to those around them. “The individuals with Delta have higher viral loads, so the virus is not only more transmissible, but there is more of it in each cough or sneeze to try and get into your cells,” Nachman explained.
From this study, it is suggested that Delta replicates faster based on the observation that the viral load from Delta was ~1000x higher than that of other strains by the time an infected subject tested positive:
Compared to the 19A/19 B strains, the relative viral loads in the Delta variant infections (62 cases, Ct value 24.00 (IQR 19.00~29.00) for ORF1ab gene) were 1260 times higher than the 19A/19B strains infections (63 cases, Ct value 34.31 (IQR 31.00~36.00) for ORF1ab gene) on the day when viruses were first detected. Considering the daily testing performed for the central isolated subjects since the beginning of quarantine, the higher within host growth rate of the Delta variant was proposed, which led to the higher viral loads on the time points when viral nucleotides excess the PCR detection threshold. [...] For the Delta variant infections, 80.65% samples contained >6x105 copies/mL in oropharyngeal swabs when viruses were firstly detected compared to the 19.05% samples in 19A/19B infections. These data highlight that the Delta variant could be more infectious during the early stage of the infection.
There might be multiple factors as to what else makes Delta spread faster, and spread among both the vaccinated and unvaccinated--and while we have some idea why, we don't have all the pieces just yet.
From an actual virologist:
We know that if we look at the spike protein, which is the protein the virus uses to attach to cells and start the infection process, we see that there are mutations that make that protein better at entering human cells. We can also look at the spike protein and see mutations that should reduce the ability of some of the antibodies generated by the vaccine to bind to the virus. So we think it's also finding ways to get around the immunity that we're generating in the population through vaccination.
That said, it's important to emphasize that while some vaccinated folks can still spread infection, today's vaccines are still substantially effective in preventing more serious symptoms against both variants. While we're still gathering more numbers, we're seeing vaccines offer a greater likelihood in effectiveness against variants such as Alpha, more than Delta, and an even greater likelihood in reducing serious symptoms leading to hospitalization, ongoing symptoms, or death related to COVID-19 in general.
Per the same virologist above (Andrew Pekosz at Johns Hopkins U):
Say, for example, it's a one in a million chance that a mutation will
be advantageous to the virus. If you let the virus replicate itself
900,000 times, odds are that the advantageous mutation will occur. But
if you limit the overall replication of the virus to 1,000 times, then
it's much less likely that the random advantageous mutation is going
to occur. And that's where public health interventions really help us
a lot during this pandemic—by reducing the total amount of virus
replication and therefore reducing the chances that the virus can
improve or adapt.
TLDR: More people getting vaccinated sooner rather than later substantially helps us in the race against further spread and mutation of SARS-CoV-2.