Once you've seen a Eurasian Collared Dove and a Mourning Dove at similar range, they become quite easy to tell apart. The same can't necessarily be said about watching them race across the sky.
The first thing you will notice is simply the size—Eurasian Dove are 1.5 to 2x larger than mourning dove are.
The second thing you may notice is the song they sing. Mourning dove have a familiar coo. Eurasians, on the other hand, make a raspy sounding koo-KOO-koo call that isn't nearly as pleasant on the ears.