The Difference Between the U.S. & European iPhone
Apple's iPhone redefined the smartphone market, and it is still, by volume, one of the top-selling smartphone models in circulation. Apple maintains tight control over the user experience of the iPhone, and there aren't as many customization possibilities with iPhones as there are with smartphone designs from other manufacturers. As of the iPhone 5S, the U.S. iPhone and iPhones sold in Europe and Asia are nearly identical. While Apple may change the following information with the launch of the iPhone 6, it is unlikely that the hardware differences on the radios will change.
The iPhone comes in two major varieties: those that use CDMA for their 3G data connection (Sprint and Verizon) and those that use the GSM standard (AT&T and T-Mobile). Prior to the rollout of LTE, the standard advice for international travelers was to take the GSM model phone, usually AT&Ts, because the vast majority of carriers used GSM phone networks outside the U.S. Apple made all of their GSM-phones quad-band phones with the iPhone 4, back in 2011, which eliminated the primary hardware distinction between the European and American models.
While quad-band GSM phones got rid of the distinctions between GSM phones, LTE 4G networking has 25 radio frequency bands. Due to limitations on radio antennas and battery life, no one LTE-capable phone can currently use all 26 bands, and the major differences now between iPhone models involve the LTE bands the phone can accept. Both the AT&T and Verizon models use LTE bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20 and 25. The model sold by T-Mobile uses LTE bands 1, 2, 3, 4, 5, 8, 13, 17, 18, 19, 20, 25 and 26, while the European GSM model uses LTE bands 1, 2, 3, 5, 7, 8, and 20.
Apple also sells iPhones tailored for the Asian market. One has no LTE option at all and is sold unlocked in China; the other model uses the 3 TD-LTE bands 38, 39, 40 and 7 FDD-LTE bands 1, 2, 3, 5, 7, 8, and 20 which is also compatible with European GSM phones.
Notable Capability Differences
The single largest source of international iPhone incompatibility is the U.S. models' lack of support for LTE band 7. LTE band 7 was used exclusively in many countries with LTE networks, and is supported on all the European and Asian iPhone versions. It is still the dominant frequency band for data in Russia, Eastern Europe, Scandinavia, Colombia, Brazil, Australia and Saudi Arabia.
Locked and Unlocked iPhones
Unlocked cell phones allow you to put the GSM card of your choice into the phone; this is a handy way to move your phone between different network carriers, and is a common trick used by international travelers. Some carriers sell the iPhone in a locked condition, preventing you from doing this. Apple does sell unlocked iPhones, but they don't, in the U.S., have the carrier subsidy that brings the price down. In Europe, unlocked phones are much more common, as is switching between carriers.
While accessory compatibility may change between iPhone generations, accessories for an American iPhone will work for a European version of the same models. The cases and external connectors are identical.