The reason I'm inclined to dismiss this difference is that Russell's advice, if you take it at all seriously, also requires a great deal of introspection. He recommends that you develop as accurate a picture of yourself, your motives, and your abilities as possible. He recommends watching for "foolish thoughts or feelings" coming up from the subconscious, so that you can "pull them up by the roots, examine them, and reject them." [page 83] Russell's advice to focus your attention outward seems mostly to be advice against dwelling on topics such as: how unfortunate you are, how sinful you are, how unfairly the world treats you, and so on. I doubt the Dalai Lama disagrees with Russell on these topics.
A good example of this Western attitude comes from the introduction science fiction author Ray Bradbury wrote to The Stories of Ray Bradbury. "I learned that I was right and everyone else was wrong when I was nine. Buck Rogers arrived on the scene that year, and it was instant love. I collected the daily strips, and was madness maddened by them [sic]. Friends criticized. Friends made fun. I tore up the Buck Rogers strips. For a month I walked through my fourth-grade classes, stunned and empty. One day I burst into tears, wondering what devastation had happened to me. The answer was: Buck Rogers. He was gone, and life simply wasn't worth living. The next thought was: Those are not my friends, the ones who got me to tear the strips apart and so tear my own life down the middle; those are my enemies. I went back to collecting Buck Rogers. My life has been happy ever since. For that was the beginning of my writing science fiction. Since then, I have never listened to anyone who criticized my taste in space-travel, sideshows or gorillas. When such occurs, I pack up my dinosaurs and leave the room."
I believe that Russell would agree that Bradbury was happy, but that the Dalai Lama would doubt it. Bradbury's "happiness" in this passage comes from a realization of his separateness from other people, not from a realization of connection. He is not, in this passage, achieving compassion or replacing anger and hatred with patience and tolerance. He is, on the other hand, discovering his zest.
Zest is perhaps the fundamental concept in Russell's view of happiness. I can find no similar concept in the Dalai Lama's book. He either doesn't think it is important, or he takes it for granted. Many of the obstacles to happiness that Russell discusses are blockages in zest: boredom, depression, pointlessness. Russell says little about anger or hatred. The Dalai Lama, on the other hand, talks at great length about anger and hatred, but says little about boredom and pointlessness.
The Dalai Lama, on the other hand, says little about biological drives. Obviously he believes that human beings should eat and drink and sleep, but his own life is that of a monk, celibate and childless. The second of the "Four Noble Truths" of the Buddha states: "The cause of human suffering is undoubtedly found in the thirsts of the physical body and in the illusions of worldly passion. If these thirsts and illusions are traced to their source, they are found to be rooted in the intense desires of physical instincts." (Source: The Teaching of Buddha by the Buddhist Promoting Foundation of Japan.)
The nature that the Dalai Lama rests his beliefs on is the Buddha nature, a universal spiritual presence that has infinite compassion and desires that all sentient beings escape their suffering. This Buddha nature is the underlying source of our consciousness, and is hidden from us by the illusory power of the physical world.