As Netflix grapples with fallout from historic layoffs and upheaval in its senior executive ranks, the streaming giant is facing mounting scrutiny from investors about its growth strategy, diversification plans and response to rising competition from the likes of YouTube .

But the biggest challenge confronting the Los Gatos, Calif., company may be something more fundamental: a clash between the firm’s roots as a technology-driven upstart and its ambition to be a force in Hollywood.

Increasingly, tensions have flared between the L.A.-based executives dedicated to movies and TV shows and its engineers and technologists in Silicon Valley.…