...Lets look at a key difference...
'Plymouth Argyle' had their first match in 1903, as a professional football side in the Southern league, Argyle played against West Ham United and won 2 0 in London. In their first home match they played against Northampton Town and won 2 0 infront of 4438 people.
'Exeter City' had their first match in 1904, as a non professional football side in a match ( east devon league)against 110th battery of the Royal Artillery, City managed to scrape through the match and win 2 1, this incredible feat was achieved infront of an outstanding 600 people.
'Plymouth Argyle's' first kit was Green and White
'Exeter City's' first kit was also Green and White