The events are so close in time, and so closely overlapping with other events, that disentangling them causally is probably impossible. It's also not clear that it's an either/or problem — as the meme says, "why not both?" I recently wrote a piece that touched on why this was complicated. Here's a very abbreviated list of the events at the end of the war:

• The bombing of the city of Hiroshima
• The Soviet declaration of war against the Japanese, and subsequent invasion of Manchuria
• The bombing of the city of Nagasaki
• Internal friction within the Japanese high command
• An attempted coup by junior military officers
• An offer of surrender that still maintained the status of the Emperor
• A rejection of this offer by the Americans
• An increase of American conventional bombing
• An acceptance of unconditional surrendered by the Emperor himself

Now what you want to ask is, can we imagine removing a few of those, and end up with a similar result? History doesn't really have the tools to answer a question like that. We can certainly play with ideas, and say, well, the Soviet invasion seems to have really affected both the military and the diplomatic factions of the Japanese Supreme War Council, so we certainly can't eliminate it from that list and expect it to be exactly the same. But what about, say, just Nagasaki? What about the attempted coup? The return to conventional bombing by the US? It's hard to say.

I don't think there's any way to conclude that it was just one or the other. I do think one has to emphasize both of them. But I think one also has to emphasize that it wasn't a straightforward "atomic bombs = surrender" line; it's clearly not as simple as that.

The best single book on all of this is Hasegawa's Racing the Enemy. In the conclusion, Hasegawa works through some counterfactuals. In the end, he leans towards the Soviet invasion being more important, and imagines that the war would have ended prior to US invasion with the Soviet invasion alone, and is unclear whether it would have ended prior to that based on atomic attacks alone (which would include not just the two used, but the ones that would follow). But whatever one thinks of his conclusions, the book as a whole is a great explication of the complexity of the end of the war.