Out-of-Universe: Patrick Stewart probably didn't want to permanently don a costume like the experience that Michael Dorn had to endure. Being anti-Borg may not have been a primary defining characteristic of Picard's overall character, at least when they starting showing the very next episode. In contrast, Seven's character was always intended to have lingering Borg references.
In-Universe: Picard could sense the Borg via a dream, as shown in the opening scene of Star Trek: First Contact. He also seemed to be able to listen to their communications, which led to Picard choosing how Starfleet should start to act during the remainder of the battle with the Borg Cube from that movie. So, Picard may have been de-Borgified in external appearance, but perhaps still had some lingering Borg pieces lying around.
Yes, I know the counter argument: IMBD Quotes from The Drumhead (Drumhead script,Drumhead script) where Picard says "Yes, I have completely recovered." However, Picard was making a statement to defend what was at stake, which was his credibility. As shown in the Star Trek: First Contact movie, that statement may not have been entirely true, from a Biological point of view. Even if Picard did have some remaining nanites in his blood, if his day-to-day life wasn't consciously affected, and his judgments were purely originating only from biological components, Picard's statement may have been entirely honest.