Nope: diplomatic immunity extends only to diplomats and their immediate families. As Snowden is neither, and cannot be nominated as one without the host country's approval, there's nothing stopping Russia or Cuba from detaining Snowden if they wanted to, no matter how many Elbonian diplomats he's travelling with.
As I used to tell friends in school, the best benefit was that I could kill people who annoyed me and get away with it...
Somewhat more seriously, pretty much the only perk was getting to use the Diplomats & Crew line at airport immigration. Most countries, including mine, make their diplomats pay their parking fines etc, and while I was theoretically exempted from security checks, it was virtually always faster to put your bags through the X-ray and walk through the metal detector than it was to pull rank and wait forever as the proto-TSA at the time ran around like headless chickens, looking for the manager's manager's manager needed to authorize the exception. The police treat diplomats with either thinly disguised contempt (if they ran into them a lot and knew they couldn't do much of anything) or, more commonly, puzzlement; and once again, it was usually way easier to pop the trunk and let them have a look than it is to invoke immunity and deal with the resulting confusion.
And as far as complimentary upgrades to first class and "ambassador's lounges" (wot?) do, I'm not sure what the other reply has been smoking, but it's probably not the Vienna Convention. Airlines don't give any free benefits to diplomats, they earn their status through butt-in-seat miles like the rest of us.
I used to have diplomatic immunity as well. Getting to ignore parking tickets and other fines, police officers treating you like royalty, no security checkpoints or lines at the airport (and a very comfy ambassador's lounge), complimentary upgrades to first class and not having to pay taxes (sales or otherwise) at the host country are the first that come to mind.
I was under the impression that they were flying on a privately owned plane. That does not seem to be the case, but it may be on their next leg of the journey.
I do not have direct knowledge, but I would think that if the plane is of the diplomat, then the whole plane could be considered to be territory of the nation of the diplomat, therefore the diplomat could offer him asylum while on the plane.
If I wasn't clear, I was not speaking about offering him diplomatic immunity, but rather the plane being considered as territory of the nation of the diplomat.
(Source: I used to have diplomatic immunity.)