I'm not confident I can predict what the Dalai Lama was trying to say (and I haven't read the book you referenced), but here are some examples of what "this distinction" means to me.
Vices and virtues
The Zen story "Nothing Exists"
Yamaoka Tesshu, as a young student of Zen, visited one master after another. He called upon Dokuon of Shokoku.
Desiring to show his attainment, he said: "The mind, Buddha, and sentient beings, after all, do not exist. The true nature of phenomena is emptiness. There is no realization, no delusion, no sage, no mediocrity. There is no giving and nothing to be received."
Dokuon, who was smoking quietly, said nothing. Suddenly he whacked Yamaoka with his bamboo pipe. This made the youth quite angry.
"If nothing exists," inquired Dokuon, "where did this anger come from?"
... implies that the self, the sense of self, conceit, identification (with form but also self-importance), is manifested as or associated with defilements and obstacles, fetters (for example, anger).
I guess this is what people mean when they talk about "the ego", being "egocentric", "selfish", etc.
The story may be, also, a reminder of social conventions: be polite (not boastful), don't attack people (be gentle), be skillful (appropriate interaction with the right person at the right time), etc.
I think that uprooting "defilements" is one the main aims of Buddhism, isn't that so?
Still, if there are bad deeds and afflictive emotions, there's also skillful virtue -- including generosity, for example, being a good friend, even avoiding bad friends -- which may lead to the next topic.
If you look at the five skandhas, you can't (or shouldn't) look at them and say "that's me, that's mine, that's permanent" -- a body for example, or any single/specific thought, is more-or-less transient, impermanent. You might say, "That's me", but maybe you won't want to: because clinging to impermanent things like that is (or would be) a cause of suffering.
Socially though, in society, people exist -- with names and so on. As well as the five skandhas, there are also (or one might say, "people also have") names and relationships, duties, jobs, families, skills, assets (like a house or money), etc. These too are temporary, severable -- and may be dissatisfactory, a cause of suffering if you cling to them or identify with them.
Even so, society, human life, couldn't exist with them -- without parents, for example, or without children, and so on. One of the definitions of "wrong view", in the suttas, goes,
"And how is right view the forerunner? One discerns wrong view as wrong view, and right view as right view. This is one's right view. And what is wrong view?
'There is nothing given, nothing offered, nothing sacrificed. There is no fruit or result of good or bad actions. There is no this world, no next world, no mother, no father, no spontaneously reborn beings; no brahmans or contemplatives who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.'
I think that this is an important part of the conventional self -- it's relative -- there's a "this world" because there is (or was) "another world", there's a "next world" because there's a "this world", there's a "mother" and "father" because there's a "child", etc.
"Relative" also implies "conditioned" -- A exists because B exists, A and B are related, A is conditioned by B -- and anything conditioned is impermanent (A ceases or changes when B ceases).
Although these relations exist, they're impermanent (so shouldn't be taken to be a permanent self), yet they exist (to deny they exist would be wrong view).
We're told to consider that,
‘I must be parted and separated from all I hold dear and beloved.’ …
I am the owner of my deeds and heir to my deeds. Deeds are my womb, my relative, and my refuge.
I shall be the heir of whatever deeds I do, whether good or bad.’
... where "deeds" is kamma in the above.
So perhaps "I am" (the owner) and "I have" (the fruit or result of good and bad actions), until I am able to "end kamma".
The Dhammapada says,
You yourselves should make the effort; the Tathagatas (Buddhas) only can show the way.
I guess this might be the kind of message which the Dalai Lama had in mind, when he was quoted as saying, "our own personal experiences demonstrate that we as subjects and agents of our own lives, we certainly exist".
Speaking of "refuge", I think the "island" passage from the Maha-parinibbana Sutta implies some agency or responsibility:
Therefore, Ananda, be islands unto yourselves, refuges unto yourselves, seeking no external refuge; with the Dhamma as your island, the Dhamma as your refuge, seeking no other refuge.
I think of that as a defining characteristic of Buddhism, perhaps unlike some other religions where worshippers might expect another agent (God) to be an instrument or agent of their salvation.
I think that other schools of Buddhism (e.g. Tibetan) might have slightly different doctrines, though, so, again, this is just how I understand the "distinction", based on the suttas (not the Heart Sutra nor the Dalai Lama's teachings, especially "a crucial aspect of the Buddha's teaching on emptiness").
By the way, even though this appears to emphasise "you" (or "self") as agent, it's also possible to describe the process (of following and/or not following the way, the dhamma) using a non-self kind of metaphor: as important as the conventional self is, it doesn't or needn't override a non-self view.
If you'll forgive my introducing it, here is one of the parables of Jesus (note Christian, not Buddhist):
"Listen! A farmer went out to plant some seeds. As he scattered them across his field, some seeds fell on a footpath, and the birds came and ate them. Other seeds fell on shallow soil with underlying rock. The seeds sprouted quickly because the soil was shallow. But the plants soon wilted under the hot sun, and since they didn’t have deep roots, they died. Other seeds fell among thorns that grew up and choked out the tender plants. Still other seeds fell on fertile soil, and they produced a crop that was thirty, sixty, and even a hundred times as much as had been planted! Anyone with ears to hear should listen and understand."
The parable/metaphor there is:
- Farmer "scattering seeds" = "giving dhamma talks"
- "Unfertile ground" = "some people hear the talk but the dhamma doesn't take root"
- "Fertile ground" = "some people hear the talk, and practice appropriately as well as spread the word"
Further to that metaphor you could argue that (notwithstanding what I said above about being the owner of or heir to the fruit of kamma) there is still no self and maybe even no "free-will" -- there's just ground, fertile or unfertile.
Curiously the last sentence of that quote ("Anyone with ears to hear should listen and understand") is rather similar to the Buddha's "beings with little dust in their eyes":
Open are the doors to the Deathless
to those with ears.