Is there a difference between iPhone LTE band support and carrier compatibility? I have an unlocked iPhone 5s model 1533 GSM which supports LTE bands 3, 7, and 19 (800, 1800, 2600 MHz) as per this page: http://www.apple.com/iphone/LTE. These are the bands used in Spain (http://prepaid-data-sim-card.wikia.com/wiki/Spain). However, the Apple page suggests that even though the 1533 model is compatible with these bands, it is not compatible with Spanish carriers and instead you need model 1457. Note also that model 1457 is not listed as compatible with Canadian carriers even though it is compatible with the LTE bands used here (700 and 1700 MHz).

Does this mean that although the Spanish LTE frequencies are supported, the phone will not work on LTE in Spain because the carriers aren't supported on model 1533?

Or does it mean that although not officially supported, the 1533 model will work with Spanish carriers if I purchase the appropriate sim in Spain?
If Apple says you need a specific iPhone 5s for Spanish carriers, then you probably do. Even if a US iPhone supports all the LTE bands in Spain, you're just better off buying the phone made specifically for your country.