Is it possible to follow Battlestar Galactica 2003 TV series without watching 1978 original Battlestar Galactica series?
You don't have to watch both, they are not related or have any sort of continuity, but you need to watch the 2 episodes mini series before watching the 2003/2005 version.
As someone who watched and loved the 1978 series as a teen... No, you don't need to watch both.
They are actually surprisingly unrelated. They have roughly the same central concept of a fleet which is fleeing genocide at the hands of the "Cylons", but that's really about where the similarities end. The Cylons are completely different in the original show. They are The Other. Shiny robots with one evil scanning eye. You could look at any character and know instantly which side they were on. The result was kind of like a weekly space-based version of the Battle of Midway.
The new one making them look just like humans completely changes the show, adding a dimension of constant paranoia and psychological drama that was just not in there in the 70's (although as a teen boy, I'll admit it could have been in there and I would not have noticed).
The fact that some of the Cylons don't even know they're not human adds a further element.
My personal suggestion would be that if you enjoy modern dark dramas with ongoing stories (eg: Arrow), then just watch the reboot. If you'd prefer an old-school series with a lot of space dogfights where noting much changes at the end of any episode watch the old one. If you've got oodles of free time on your hands, feel free to watch both.
In a really weird way, the reboot is also a sequel series to the original galactica series. In Galactica 80, the fleet has found earth and they send a small contingent of children to live on it. The cylons also send a small contingent (no idea why)that are humaniform cylons to hunt them down.
One of the big themes of the reboot is "all this has happened before and will happen again" which might indicate that the previous series were within canon of the new. But that's really not worth thinking about. this is "brains off" TV. They don't have a plan, and neither did the writers. Take it episode by episode and don't try connecting the dots, you'll be much happier about the series for it.
I have never watched more than a couple of minutes of the 1970s version but happily watched all of the 21st Century reboot. As other have said, the 21st Century version can confuse people because it begins not with Season 1 but the Mini Series, which is possibly the best part and not to be missed, apart from showing the beginning of the story. It stands up to repeated viewing as not all later Seasons do.
Also note some later episodes such as 'Razor' were released individually but are also included in the relevant series.
Tricia Helfer is especially brilliant and sad she has not had other more prominent and varied roles since, unless I missed them.
Note two interesting prequels were made later with different casts, a TV movie set early in the Cylon War called Blood & Chrome and a series set before the Cylon War called Caprica.
There was another called sequel 'The Plan' featuring the original cast but of more mixed interest.