'背包' is much, much commoner than '揹包':
- '背包' + '臺灣': 03,320,000 results.
- '揹包' + '臺灣': 00,269,000 results.
- '背包' + '台灣': 15,800,000 results.
- '揹包' + '台灣': 00,672,000 results.
This agrees with my experience. Here is more evidence:
- '背包' + '購' + '價' + 'tw': 3,640,000 results.
- '揹包' + '購' + '價' + 'tw': 0,170,000 results.
- '背包' + '選' + '裡' + 'tw': 1,900,000 results.
- '揹包' + '選' + '裡' + 'tw': 0,181,000 results.
A possible explanation is that wearing a backpack doesn't really need use of hands.
BONUS:
In the following examples, '揹' is no less popular:
- '背著洋娃娃' + 'tw': 6,590 results.
- '揹著洋娃娃' + 'tw': 9,510 results.
- '背著年邁的' + 'tw': 1,260 results.
- '揹著年邁的' + 'tw': 1,880 results.
- '背小孩' + '媽媽' + 'tw': 03,920 results.
- '揹小孩' + '媽媽' + 'tw': 06,940 results.
- '背嬰兒' + '媽媽' + 'tw': 10,700 results.
- '揹嬰兒' + '媽媽' + 'tw': 13,000 results. (e.g., 英國 WMM 無敵揹巾)
In the following examples, '背' is generally, very, very strongly preferred:
- '背負' + '責任' + 'tw': 1,140,000 results.
- '揹負' + '責任' + 'tw': 0,080,600 results.
- '背負' + '期待' + 'tw': 1,120,000 results.
- '揹負' + '期待' + 'tw': 0,072,700 results.
- '背負' + '重擔' + 'tw': 0,104,000 results.
- '揹負' + '重擔' + 'tw': 0,014,900 results.
- '背負' + '期許' + 'tw': 0,102,000 results.
- '揹負' + '期許' + 'tw': 0,008,460 results. (e.g., 為什麼華人社會的小孩都要背負期待和壓力長大)
I hope that these comparisons help you understand how we Taiwanese choose between the two characters.