tp-link xxxHP series is stand for HIGH POWER, and it refers to the machine's wireless transmission power, rated by mW (miliwatt) or watt.

Antenna is another thing, rated by gain (dbi). so 'high power' and 'high gain' basically 2 things. last time most home router with 100mW (0.1watt) only then pair with high gain antenna 5dbi to 9dbi max (the limit reached for indoor antenna). Thus product always labelled with wording of 'high gain', not using the term 'high power'.

early 841hp version is rated at 800mW (0.8watt), the newer version should be at 1000mW (1watt) same as asus.

technically,

8dbi antenna is 7.x dbi effect (seven point sth)
9dbi antenna is 8.x dbi effect (eight point sth)
no sharp 8dbi or 9dbi

for omni-directional type antenna, higher dbi means wider horizontal line coverage size
their coverage model is a donut shape, not a ball shape. higher dbi means the donut will get flatter. (from a donut turn into a pizza...)

to see obvious difference, the gap should be at least 3dbi... e.g jump from 5dbi to 9dbi
8dbi to 9dbi seems no major overhaul, but should have little gain at further distance spot