How is Newton's first law of motion different from Galileo's law of inertia? If the two are the same, then why is the first law named after Newton? Galileo's law of inertia (at least what I've learned) is

"A body moving with constant velocity will continue to move in this path in the absence of external forces".

And Newton's first law says

"A body moving with constant velocity will continue to move with the same velocity provided that the net force acting on the body is 0.

So how are these two laws different? Both laws are telling us that a body in uniform motion will continue to move in uniform motion until the net force on the body is 0.
And if they are the same thing, then why isn't Galileo given credit for the above law? We call the law "Newton's first law of motion" when in fact Galileo discovered the above law through his experiments with inclined planes.
 A: I believe that Galileo did not state clearly that the natural motion of a body would always be in a straight line. In some circumstances (such as for an object thrown horizontally and at the right speed), the natural motion would be at constant speed in a circle around the Earth. [In the second day of the Dialogue concerning the two Principal World Systems he talks about "a ship moving over a calm sea" being "one of those movables which courses over a surface that is tilted neither up nor down, and if all external and accidental obstacles were removed would be disposed to move incessantly and uniformly from an impulse once received." Earlier he has stated that for a surface to be neither upwards nor downwards "all parts must be equally distant from the centre." (Stillman Drake translation; my italics).]
Descartes, though, did state, before Newton, that a body's natural motion was in a straight line at constant speed.
Remember that the full concept of force, and gravitational force in particular, came later – largely due to Newton. Therefore neither Galileo nor Descartes could include the general proviso about forces, such as "in the absence of external forces".
A: They are the same. As to why we know it by Newton's name today, that is a question more suitable for the History of Science and Math Stack Exchange.
I actually don't like the second formulation as quoted. I think the original (translated) is better

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

I prefer something closer to Newton's original because it seems confusing to introduce the concept of net forces in the first law when they won't be defined until the second law.
A: In the interest of motivating this convention, rather than just mulling its history (we're not on hsm.se yet):
You can call it Galileo's law of inertia if you like, but it makes sense to call the collection of laws Newton's, just as Gauss's law (and a magnetic counterpart), Ampère's law and Faraday's law are respectively Maxwell's equations. In both cases, realizing the mathematical implications of assembling what were previously stray observations is what earns nominal recognition.
Having said that, Maxwell's equations aren't numbered. (This is not only unsurprising, but helpful: he thought of the collection as eight equations, they're normally taught today as four or relativistically two, of which only one is non-tautological, if we don't include magnetic monopoles.) Newton's laws are in that respect more analogous to Kepler's, which probably has some historical explanation beyond the present scope. Insofar as the numbering reflects chronology, Galileo's prior work may explain why the "first two" laws are separate, but that's just a conjecture on my part. However, when you learn general relativity, you immediately become grateful Newton extracted out the special case $\vec{F}=\vec{0}$.
A: This is not uncommon in the world of physics. Different scientists might discover certain parts of a theory and some other scientist might come later and be able to incorporate all the previous known bits of knowledge into a more complete, full theory.
Even in case of relativity, many parts of it were discovered by other scientists/mathematicians before Einstein. But Einstein was the first one to take all of those, add his insight and combine all of it into a whole, consistent, complete theory that provided a much better explanation than any theory before that.
In case of Newton as well, although Galileo's law of inertia, predated Newton, Newton was the first to combine that with Newton's 2nd and 3rd law, combined with his brilliant work in calculus and was able to develop a complete theory of mechanics that was better than any other theory till then.
For what it's worth, I have seen many physics books where chapters on Newton's laws give honorable mention to the fact that Galileo came up with his law of inertia, years before Newton.
