Often times, one of the main differences between Android phone models is the supported frequency bands. This can be because they simply use different antennas for different bands, or because they support completely different network standards (e.g. CDMA and GSM versions). It is not uncommon for devices to be released with completely different names on different networks, in fact, when they are essentially the same device. For example, the original Galaxy S had a number of named variants on different carriers: the Captivate, the Vibrant, and the Fascinate.
There can also be other hardware differences between variants as well. In this case specifically, the i9500 international version of the GS4 doesn't have an LTE radio, although the i9505/i9506 does. In the GS4's case, this is because they are built with entirely different systems on a chip: the i9500 uses an Exynos whereas the others use Snapdragons. Significant hardware differences like this are, from what I've seen, not generally as common (although I have no hard numbers), though some changes to less essential things like the camera or NFC support could exist as well1. The spec sheets for each version should outline this information, though, so it is worth checking since there is no real way to "generalize" these differences.
The international versions will work on AT&T and T-Mobile USA's 2G networks, as well as AT&T's 3G and certain T-Mobile 3G areas (ones using the 850 MHz band), but won't support LTE on either. They will not work at all on Verizon, because they don't support CDMA networks nor do they support Verizon's LTE band. Even if they did, Verizon will typically not activate non-branded phones on their network. If you're going to use Verizon, you need to buy a Verizon phone. Can I use my device on a different carrier? has more detailed information on how to make these determinations.
A ROM for one phone will not typically work on other variants, or if it does it will be somewhat broken. For example, running a ROM for the GSM Galaxy Nexus (maguro) on a CDMA variant (like toro) would often work since the hardware was largely the same, but you couldn't make phone calls or access 3G data or the like. In this specific scenario, I would not expect a ROM for the international GS4 to run on another variant at all. They use completely different systems on a chip.
In most cases, the only real "con", as such, to owning an unlocked/off-network phone is that you cannot get official support for the device through your carrier. They will often be willing to provide some kind of equipment protection plan or insurance, but if you go into a store or call customer service for technical support they will not be able to help you.
1With Samsung it may be slightly more common, because of the nature of the situation. They designed the Exynos chipset, which is why they use it in their devices in place of Qualcomm's Snapdragon, but they did not have an Exynos SoC that supported LTE at the time. Thus, in markets where they intend to support LTE, they sometimes go with entirely different chips (like Snapdragons).