byams dang snying rje
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The difference is that byams is feeling happy about others, sentient beings, and wish them all the good. snying rje is feeling compassionate about others, sentient beings, and wish them to be free from any sufferings.
This is where two famous bodhisattvas have their names from, Maitreya, is byams pa, the Loving One. Avalokiteshvara has many names in Tibetan, of which one is snying rje'i lha, Lord of Compassion.
To conclude, these two are also part of the four immeasurables, tshad med bzhi: byams pa, snying rje, and also dga' ba, sympathetic joy (in Sanskrit mudita), and btang snyoms, equanimity (in Sanskrit upeksha).