With the older saw, the first thing to suspect is the spark plug, because it has the right symptoms: hard to cold start, not so hard to restart. However if it had been standing a long time when it was hard to cold start, stale fuel in the carburetor is also a possible problem. Unless you deliberately run the carburetor dry before you store it, the petrol can dry up leaving oil, which won't go through the jet under venturi suction alone. So, you might try a simple service: clean or replace the air cleaner element, replace the spark plug if there is a possibility it has run 15 hours or more, run the saw dry of fuel, let it cool off for a couple of days, then see if it starts better. If not, you will need to do a full service, tuning the high and low speed mixture controls, and probably including overhauling the carburetor (replacing both diaphragms) if the saw is more than 5 years old.
Because the new saw had the hard starting characteristic immediately from new, it is more difficult - I had a little Sachs Dolmar saw that had the same problem, and I finally solved it by donating it to a local service club. I think all you can do is approach it from first principles: check the spark, then try priming it with a spoonful of fuel directly through the spark plug hole to see if it will fire when you do that. If it will fire when primed, but not otherwise, you will have to investigate the fuel system: tank venting, fuel lines clear and not split, etc, then tune the carburetor and possibly overhaul it. Chances are it is a defective component or adjustment: bad spark plug, split fuel suction hose, bad in-tank fuel pickup filter, stuck tank vent, kinked hose, foreign body or burr in the carburetor, etc. Start with the ignition then the fuel tank and work forward, leaving the carburetor for last: many people who "fix" hard starting chainsaws by attacking the carburetor make things a lot worse.
Some chainsaws require oddball starting techniques, and if you don't happen to find out what yours likes, life can be miserable.
