If one takes a slightly more expansive view of the word "saving" to include most forms of durable asset accumulation, I think the reason some do and most don't is a matter of a few factors, I will include the three that seem obvious to me:
Education
Most schools in the US where I live do not offer personal finance courses, and even when they do, there is no opportunity for a student to practice good financial habits in that classroom setting. I think a simple assignment that required students to track every penny that they spend over the period of a few months would help them open their eyes to how much money is spent on trivial things that they don't need. Perhaps this would be more effective in a university setting where the students are usually away from home and therefore more responsible for the spending that occurs on their own behalf.
Beyond simple education about personal finances, most people have no clue how the various financial markets work. If they understood, they would not allow inflation to eat away at their savings, but that's a separate topic from why people do not save.
Culture
Since much of the education above isn't happening, children get their primary financial education from their parents. This means that those who are wealthy teach their children how to be wealthy, and those who are poor pass on their habits to children who often also end up poor. Erroneous ideas about consumption vs. investment and its economic effects also causes some bad policy encouraging people to live beyond their means and use credit unwisely, but if you live in a country where the average person expects to eat out regularly and trade in their automobiles as soon as they experienced their highest rate of depreciation, it can be hard to recognize bad financial behavior for what it is. Collective savings rates reflect a lot of individuals who are emulating each other's bad behavior.
Discipline
Even when someone is educated about finances, they may not establish good habits of budgeting regularly, tracking spending, and setting financial goals. For me, it helps to be married to someone who has similar financial goals, because we budget monthly and any major purchases (over $100 or so) must be agreed upon at the beginning of the month (with obvious exceptions for emergencies). This eliminates any impulsive spending, which is probably 90% of the battle for me. Some people do not need to account to someone else in order to spend wisely, but everyone should find a system that works for them and helps them to maintain some financial discipline.