The difference is in speech level. There are seven speech levels in Korean, and each level has its own unique set of verb endings which are used to indicate the level of formality of a situation "입니다" is Hapsyo-che(합쇼체) It is used between strangers at the start of a conversation among male co-workers by TV announcers to customers "이에요/예요" is Haeyo-che(해요체) It is used mainly in Korean phrasebooks for foreigners between strangers, especially those older or of equal age between female co-workers or friends by younger speakers as a less old-fashioned alternative to the hao-che. by men and women in Seoul as a less formal alternative to the hapsyo-che. Refer to http://en.wikipedia.org/wiki/Korean_speech_levels

First, two types are polite expression Second, 입니다 is 'is' in 'she is beautiful.' And 이에요 is 'is' in 'is she beautiful?'