Reading the transcripts of the Obama and Putin speeches reveals what we should describe as two conflicting theories of destabilization. According to Obama, "repressive" regimes like Assad's effectively destabilize their countries before anyone takes up arms against them. The American President extends a presumption of legitimacy to uprisings against repressive regimes, so long as the insurgents respect human rights and civil liberty, that Putin most likely does not. The Russian President is more likely to attribute uprisings to meddling from the outside, particularly in Ukraine, whose Maidan uprising he described as a "military coup," but also in Syria, where American "social experimentation" and Sunni extremism actively destabilized a situation Putin presumably presumed stable.
Obama summarized his position best here:
Just as force alone cannot impose order internationally, I believe in my core that repression cannot forge the social cohesion for nations to succeed. The history of the last two decades proves that in today’s world, dictatorships are unstable. The strongmen of today become the spark of revolution tomorrow. You can jail your opponents, but you can’t imprison ideas. You can try to control access to information, but you cannot turn a lie into truth. It is not a conspiracy of U.S.-backed NGOs that expose corruption and raise the expectations of people around the globe; it’s technology, social media, and the irreducible desire of people everywhere to make their own choices about how they are governed.