The answer is complicated for two reasons: first, the digraph ει actually stood for two different sounds; and second, both these sounds and the sound of η later fell together into a single vowel sound. I'll describe the phonetic values of these symbols at the stage when the three different pronunciations were still distinct.
η stood for a long open-mid front unrounded vowel, IPA [ɛ:]. English has the short version of this vowel, [ɛ], in words like bed or pet. It is specifically "open-mid" rather than just "mid" because it is pronounced with a more open mouth than, for example, the e sound of a language like Spanish.
ει, as stated above, had two different pronunciations:
- Just as its spelling suggests, it could stand for a diphthong [ei], like the a in English ace, to use your example. (Your textbook is wrong about eta, which did not stand for this sound.) This is known as the "genuine diphthong" pronunciation of ει.
- But it could also stand for a long close-mid front unrounded vowel. This is the more close version (i.e. pronounced with a less open mouth) of the sound described for eta above. Standard English doesn't have this sound. Some other languages, though, make a contrast between the close-mid and open-mid front vowels: e.g. in French, parlais has the open-mid, parlé the close-mid sound. This is known as the "spurious diphthong" pronunciation of ει, since it looks like a diphthong but isn't.
Whether ει has the genuine or spurious pronunciation is unpredictable and depends on the specific word. The reason ει was used for both these sounds is that already early on in Classical Greek, the two were merging into one. Later on in the koine period, η too began to merge with them, and eventually all three ended up being pronounced [i], as they are in Modern Greek.