The Mk2 DB6 has a more attractive, pleated leather seat pattern (carried forward to the subsequent DBS model) and flared wheel arches to accomodate wider wheels and tires. The standard 3X SU carb and 3X Weber 'Vantage' engine from the Mk1 was carried over, virtually unchanged; a mechanical "AE Brico" fuel injection option was also offered with the introduction of the Mk2. A Mk2 is much rarer, and more desirable as the most refined of the range which began with the DB4, and therefore more valuable.
Running costs on these cars can be very modest if you find a well-sorted example to start with. Over-engineered and very robust. Watch for rust in the chassis, low compression or signs of overheating. A good one is a joy, and much cheaper to run than a V12 Ferrari.
Prices are all over the map, dependent upon spec and condition. As with Ferraris, buy the best you can afford; rusty frames, engine rebuilds and all the little details will put you over the cost of a good one to start with.
Generally speaking, I would add 1/3 more for Mk2. The most desirable of all, a factory Vantage-spec LHD 5-spd example (of which only a handful were built) could command double that of a standard spec Mk1.
I have an excellent, sorted, standard Mk1 (5-speed) which I would sell for USD$125K, a rational price for an excellent car in today's market - to give you an idea.
I would disagree with some of the advice given in this thread.
Although DB5 prices have been talked up by owners who overpaid in the first place and dealers trying to make a thick profit, cars that are actually changing hands privately and at auction don't show the same values. The last two DB5s I saw go at auction this year went for £28K and £40K respectively. In fact, some DB6s at the same auction went for much the same money - although they were in better condition.
I would suggest that the "James Bond" premium of the DB5 over the vastly superior DB6 is about 30% now.
Astons always carry a premium for rarity, which is another reason the DB5 and DB6 Mk2 is more expensive. After a DB6 the DB5 feels awful!
As to differences between the Mk2 and Mk1 DB6 the jury really is out on the "improvements". The interior is like the DBS one and is very "1970s", some feel that this is out of keeping with the "1960s"ness of the rest of the car.
Most Mk2s are fitted with power steering - which carries a premium of its own.
The worse aspect of the car is that it has the wheels and hubs of the DBS - although this did lead to the slightly flared wheelarches which look very nice. The bigger hubs and wheels add a lot of sprung weight which leads to far worse handling and poor suspension reaction over a rough surface. They tend to feel very bumpy indeed which makes the comment about it being "refined" an odd one. The Mk2 is the least refined and certainly the heaviest of the DB6.
The other problem is that the increase in weight due to the interior, wheels, and hubs blunts the performance somewhat. The DB6 Mk1 has better performance than a DB5 due to superior aerodynamics from the DP214 Le Mans car at a weight penalty of only 7Kgs. The extra 150Kg of the Mk2 is not good.
I don't believe any cars are left with injection. If one did come up for sale it would be very expensive indeed.
The most expensive DBs ever are the DB6 Volantes. If you have to ask the price, you can't afford one!
As to running costs, I've owned a lot of supercars in my time, and they are much cheaper than Ferraris - mainly because a lot of the parts come form commonly available cars and the rest is easily available from the factory. The last engine rebuild I did only cost around £200 in parts. Not sure I could do that with a Ferrari of any vintage!