Man, Bourne is quite unlike Bond. All they really have in common is a setting of covert action.
Bourne influenced the new Bond in several ways. The only time I remember Bond getting into actual hand-to-hand fights before CR was maybe a scene in Goldeneye. Before that it was mostly gunplay in tuxes, gadgets and maybe a poorly choreographed wrestling match. But the action sequences are beside the point, really. I'd chalk those up to prevailing action movie trends than actual specific aping.
What Bourne had that breathed life into the spy genre was a semblance of reality. Bond was all magical realism and wish fulfillment. Wouldn't it be cool if spies were actually dashing high-society douchebags with expensive toys, instead of the pencil-pushing bureaucrats and, to a lesser extent, military men that they actually are?
Bourne took the spy movie away from the glossy save-the-world-from-a-eurotrash-villain antics and brought it closer to what reality would look like were it a movie. Evil geniuses with death rays and alopecia intent on destroying the world? Silly. A highly trained wetworks agent on the run from a gov't with its hands dirty? Slightly less implausible, but it felt plausible in the post 9/11 world. It's pertinent to point out that the Bourne books are definitely influenced by Bond, but they're airport paperbacks and inferior in pretty much every way to the movie versions. Bourne was in a lot of ways an inversion of Bond. Bourne is a professional soldier who improvises under pressure. Bond is a suave, quippy jetsetter who always knows what to do. Bourne steals cars, uses public transport and hides out in slums. Bond drives a paid-for Aston Martin (which he casually trashes) and stays at beachfront resorts under his own name. Bourne opposes his handlers, shady bureaucrats who have committed simple crimes that actually are carried out by covert agencies in the name of national interest. Bond opposes rich villains with needlessly elaborate plans who are evil for ambiguous reasons.
When Casino Royale opened with grainy digital footage of Daniel Craig drowning a guy in a sink, it was meant to be a signal of a reboot in more ways than one, but more to the point it was playing at being Bourne. They couldn't keep that facade up forever, Bond has expectations to live up to, and with QoS you can see them attempt to have it both ways, having a dedicated professional soldier who also happens to be a high society douchebag, who drives Aston Martins but fights with his bare knuckles, who puts on impeccably tailored suits before dirtying them up real good, who chafes against his agency but always for the right reasons in the end, a cad who is also deeply romantic. Really it's just Bond, minus the gadgetry, with a new coat of paint. It's only going to take a few movies before it's as innocuous and tiring as it ever was.