Among other things, in the 100 years since the lore progressed from where it stood in 3rd edition the population of surface dwelling drow went from scattered populations of small to medium size that kept generally out of sight to populations of about the same size that are rather more public than they used to be.

It sounds like Eilistraee was gone for some decades and during that time her worshippers were supported by various Seldarine and human gods and Corellon undid the curse that declared the descendants of Ilythir and Miyeritar (the tribes of Dark Elves that became Drow in Faerun) who followed Eilistraee as purified. When Eilistraee reappeared she was once again Seldarine instead of Dark Seldarine, which honestly, personal thought, Sounds like Lloth kicked her out of the small realm she had in the Demonweb to make it harder for her to reach out to and free drow from Lloth.

The progression of surface-drow being more known started in 2nd edition in 1991 with the Drow of the Underdark release,

Also, Vhaerun's worship has been growing...he was also gone for a brief time, apparently after a conflict with Eilistraee though when they both came back they had a new found respect for each other....and I'm not sure what happened with his followers while he was gone...

Anyway, Vhaerun worshippers are often shady merchants and mercenaries but they work with everybody...and they hate Lloth.

Stand out public drow personalities that are not Lloth servants include:

Qilue Veladorn, one of the Seven Sisters, Chosen of Mystra and Eilistraee
Liriel Baenre, priestess of Mystra. Witch of Rashemen
Drizzt Do'Urden, ranger of Mielikki
Jarlaxle Baenre, mercenary, captain of Bregan D'aerth mercenary captain
Zhai, rogue, father was a wood elf, raised among wood elves, one of three playable protagonist in Demon Stone

Qilue and Jarlaxle are both leaders of large groups of non-Lloth drow that frequently interact with non-drow in a peaceful way. They're also representative of Eilistraee adn Vhaerun respectively. (Though QIlue is currently dead, though apparently Eilistraee hasn't given up on Qilue returning to mortal life yet)

Liriel has integrated herself into a human culture. Zhai is shown being welcomed warmly by the wood elves of her home village toward the end of the game. Drizzt sort of does the lone wanderer thing but has strong ties to Silverymoon and Mithril Hall.

If you play drow in NWN2 you get a lot of initial terrified responses but it is relatively easy for the PC to convince people they're not a monster.

Reaction-wise, surface drow in Faerun get somewhat more raised eyebrows than tieflings and half-orcs. In heavily cosmopolitan settings like Waterdeep or open-hearted like Silverymoon they're likely to get less social static than other places. Baldur's Gate has a lot of shady areas, so there's probably a good market of Vhaerun worshipers and mercenaries there.

That said, some level of concern would feel appropriate to a drow PC.