No, there isn't. The milky way is the name given to the band of stars we see due to the flattened disk-like characteristic shape of our galaxy. We can call our galaxy the milky way galaxy.
There isn't one. really. If there is a difference, you could say that the Milky Way is the milky band of light as seen from Earth, whereas the Milky Way Galaxy is the whole galaxy, which includes...
In relative terms the Milky Way Galaxy is quite a stable galaxy within the Local Group. whereas the Comet galaxy is being ripped apart by gravitational forces of the Abel 2667 cluster.
The Milkyway and irregular galaxy are both galaxies,the only difference is the shape. The irregular is a twisted and irregular shape while the milky way is a spiral kind of.