Not all groups accept posts from pages. That depends on each group. I would guess that an advantage posting from your page would be that group members or the public might "like" your page and then follow you.
If you have any reactions or likes after posting to a group from your page, you can invite those people to like your page. (Theoretically that's a positive, but pages are apparently doing badly these days overall. Regardless, I post solely from my page these days.)
Also FB says, you shouldn't do business on your personal profile, the pages are for that, so probably if you are selling heavily from your profile, you can be penalised one day? I am not sure of this, but I read something like this somewhere. So I dont post directly to my profile links from faa and aw, but I tag myself in page posts.
I keep my profile for my family and friends, so I dont accept friend requests from everybody who like my paintings, but I allow followers, and I tag myself on every post on my page, like "by @Dora Hathazi Mendes" at the end of the description, so I can make them appear on my personal timeline, allowing in the activity thing. Its mean from wherever I post to the groups, if they click on my head in the group, or on my page logo, they will see more or less the same thing both places, without creating business posts directly to my personal profile, but only to my page
I prefer to post everything from my page to groups, not only because what JW said and increasong follower this way, but because on my profile I have only around 470 followers, on my page I have 5600 followers, and if they land from the group to my page instead on my profile, it looks much better, also I have there the shopping thing from FAA, and more information about me, and about my paintings, read reviews from real people purchased from me something, than on my personal profile nothing of this, only a link to my aw.