They are quite close, Real Alcázar being on the edge of the beautiful but very touristy Santa Cruz district while Plaza de Espaná and the large Parque María Luisa are a few hundred meters away and not that central.

Here are some hotels around Alcázar with guest reviews etc. http://www.booking.com/landmark/es/alcazar-3.html?label=gog235jc-city-en-es-sevilla-unspec-no-com;sid=84f095274324473fdc984553bbd55d0d;dcid=1;inac=1&lang=en-us

If you want to stay very central and close to all the most obvious sights, I would rather recommend the much more local feel Arenal area just west and north-west of the cathedral and the Reales Alcazares etc.

Adriano Boutique is a good option in the Arenal area, five minutes walking to the cathedral. http://www.booking.com/hotel/es/adriano-sevilla.en.html

Living-Sevilla Maestranza is an apartment option in the same area and gets raving reviews. http://www.booking.com/hotel/es/living-sevilla-maestranza.en.html?aid=311984;label=living-sevilla-maestranza-*P7kGHzB3gUKk*aRK6CrmgS14880289753;ws=&gclid=CJuasvC2v7QCFTJ2cAodUVcATQ

Have heard nice things about both Apartmentos Murillo and Hotel Murillo in the midst of the Santa Cruz district.
http://www.booking.com/hotel/es/apartamentos-murillo.en.html
http://www.booking.com/hotel/es/murillo.en.html

We love the Hotel Amadeus in Seville. It is small, charming and thestaff is very helpful and friendly. The hotel has a roof terrace where a delicious breakfast is served. There are great views from the terrace including a wonderful view of the cathedral.