I updated my answer thanks to the comment from @Ben. I didn't know about the official tie-in comics that reveal a bit more information about the suits. My old answer was from on-screen info only.
All in all we have seen three War Machine suits so far in the cinematic universe.
- Mark I (Iron Man 2)
- Mark II/Iron Patriot (After Iron Man 3 up to Civil War)
- Mark III (Civil War)
War Machine MK I
The first War Machine is actually the Iron Man Suit MK II.
So in a sense they are identical. We can clearly see this in Iron Man 2 when Rhodey goes into the garage to don a suit. We see the rebuild MK I and MK II & III from the first movie in the "gallery". Rhodey then dons the unpainted MK II to fight Tony. After the fight he takes the suit with him and it's retrofitted with weapons by Hammer, turning it into War Machine.
The MK II is a prototype suit however. The first one Tony built after he returned home and the one he used for flight testing. Aside from the repulsors it lacks all of the advanced weapons Tony used in the MK III and IV suits. It also still had the icing problem(1), keeping it from reaching high altitudes.
War Machine MK II
After the events of Iron Man 2, Tony took the MK II armor back from Rhodey. He undid all the changes made by Hammer which explains why the armor is seen again in his garage in Iron Man 3. Instead he gave Rhodey a new suit, the War Machine MK II, which was also briefly known as Iron Patriot.
This suit contains the essential improvements Tony made up to this point. It was also specifically designed for combat. In this suit most of the weapons are concealed, just like in the Iron Man suits. One exception is the trademark shoulder gun.
War Machine MK III
The MK III is an upgraded suit that Rhodes uses after he becomes an Avenger and that he uses during Civil War. It mainly features some weapon upgrades like a melee weapon and a sonic cannon.
One major difference between the War Machine and Iron Man suits seems to be the AI support. As far as I can recall War Machine is never shown getting any sort of AI feedback. The suit is also never shown to act autonomously, like the Iron Man suits can in later generations. You really have to wonder how he controls the suits many functions without help. Tony always looks like he could hardly get anything done without the help of Jarvis or Friday.
As for the weapons, while War Machine features the more obvious weapons like the shoulder gun, Iron Man actually has the more advanced arsenal, including his powerful laser and several guided weapons systems.
Even the updated War Machine MK II and III armors are not more powerful than their Iron Man counterparts, even though they feature updated repulsors and similar miniaturized weapons.
So there is really no reason for Tony to use War Machine instead of Iron Man. The War Machine suits are basically just modified snapshots of Tony's progress with his own suits. He did after all create more than 40 suits with various specializations and improvements that suit his own style much better than War Machine.
1: According to the Wiki, I don't think this was somehow confirmed onscreen.