-The New Avengers breaks many of the traditional rules of the original Avengers such as including uniformed policemen, blood visibly on display and Steed using firearms in episodes such as 'Dead Men are Dangerous', 'The Gladiators', 'Hostage' and 'Trap'. Steed also keeps an antique Colt 45 from his cowboy days in his house. The lower classes are now portrayed including tramps and the central conflict is explicitly the Cold War between NATO and the Warsaw Pact rather than some unnamed 'foreign power'. In many ways the series is seen as bridging the gap between the pure fantasy of 'The Avengers' and the gritty realism of creator Brian Clemens next project 'The Professionals'.Again, as above, Joxerlives excellently describes the original 'Avengers' as per 1965-69. The 'New Avengers' episodes certainly differ from that era of the show at times, although there are still plenty of fantastical episodes - Hitler being revived, giant rats, a third Cybernauts story, for example.
However, the inclusion of uniformed police and Steed shooting a gun do tie in with the earliest seasons of 'The Avengers', 1961-1964. Many prefer to see the later series as a completely seperate entity, although many others accept the clear line of continuity - right down to the previously mentioned Cybernauts returning, and also a brief return for Emma Peel (Diana Rigg), courtesy of reused footage from the 1960s.
The show, like other long-standing shows or film series, manages to survive by re-inventing itself to a greater or lesser agree, depending on current audience preference or the wishes of a new, incoming Producer (show-runner). Same goes with the likes of Doctor Who or James Bond.Producer Brian Clemens summed up the difference saying "The Avengers" was a essentially a comedy with dramatic overtones, while "The New Avengers" was more of a drama with comedic overtones.
-Steed doesn't habitually use a gun because Patrick McNee had an antipathy to them based on his wartime combat experiences as a motor torpedo boat commander in the Royal Navy. However he does use one to free Tara in 'Game' , to bluff the courier into giving Tara his secret instructions in 'Take Me To Your Leader' and in 'Split!' he states that he shot a Russian agent in the heart a few years previously. In 'Get-A-Way' he tells a Russian agent that his favourite gun is a Smith&Wesson Magnum. In 'The Curious Case of the Countless Clues' he is seen to be carrying one and in 'Legacy of Death' and 'Homicide and Old Lace' he uses one to kill two villains. He goes on to use firearms on many occasions during 'The New Avengers' although the only gun he keeps in his house is an antique Colt 45 revolver from his cowboy days.The above answer from Joxerlives is certainly true, and well explained, of the most popular years of 'The Avengers', that is to say the colour episodes featuring Dianna Rigg alonside Macnee. These are the episodes that first gained cult popularity in the USA. However, when the series first began in 1961, it was a much straighter espionage show, with only faint traces of the later popular quirkiness. This aspect built up gradually over time. In the first two series of the show, featuring Macnee alongside Ian Hendry (Series 1) and Honor Blackman (Series 2), there is a lot of gunplay and Steed happily exchanges fire most weeks, often in extended gun battles. This was phased out, as Joxerlives quite rightly says.