Hello, Icky. I wasn't aware of differences between Artemisia absinthium and Artemisia annua, so I briefly looked into same. I found two brands which use A-annua: Gaia Herbs and Allergy Research/NutriCology. I found seven which use A-absinthium: Allergy Research, NOW foods, Eclectic Institute, Nature's Answer, Garden Botanicals, Herb Pharm, and Nature's Apothecary.
I don't know the safety issues for Artemisia annua (both independently and when compared to Artemisia absinthium), but noted that NutriCology lists these warnings on their A-annua label, "Not indicated for pregnant or nursing women. Long term administration (greater than 2 or 3 months) should be monitored by a healthcare practitioner," while Gaia Herbs states for their A-annua, "Not to be used during pregnancy or lactation."
From the second link below, "The addictive thujone (alpha-thujone) is what gave the drink absinthe its notorious reputation...absinthe, which contains thujone, is a volatile oil noted for its narcotic effect responsible for hallucinations, psychosis, and brain damage -- a syndrome labelled 'absinthism'." Allergy Research Group touts that their "ARG Artemisinin is minimum 98.5% pure, and does not contain thujone." So it would appear that the presence (or concentration) of alpha-thujone may be the deciding factor as to the "safety" of a particular Artemisia species..? The second link clearly shows alpha-thujone being present in Artemisia annua. The last link says of Artemisia annua, "No formal safety testing of essential oils; presumed moderately toxic."