In this structure they are interchangeable, only «также» is more bookish. The emphasis should be the same, with a stress on «тоже»(«также»). The sentence means that other people eats apples, and she eats them, too.
If you mean that she eats apples, as well (in addition to other things she eats), it would look as follows:
- Она ест и яблоки тоже.
In this stucture, where яблоки are framed by и and тоже, it would be a stretch to replace тоже with также. However, if you are going all clever and bookish, you can say:
- Она также ест (и) яблоки.
On the other hand, также is used as "besides, also", including the sentence-initial positions. This is something «тоже» cannot do:
- Также она знает математику. = Also, she knows maths. ~ Она также знает (и) математику. = Она знает также и математику. = Она знает и математику тоже. (these mean a slightly different thing than the first two)
In short "me too" sentences «тоже» is the only natural choice.
There is a disapproving «Тоже мне!», an expression used in colloquial speech, which does not exist with также. It is used towards a person who appears to be proud of themselselves—but you think that what they did or said is nothing to write home about. You can use it with a noun to further clarify what characteristic you deny them or the result of their behaviour:
- Тоже мне вор. Пошёл вон! ~ Master thief my ass. Bug off!
- Тоже мне умник нашёлся. Молчал бы. ~ Yeah, right, smartypants. Better shut up.
- Тоже мне откровение. ~ So enlightening! Except it wasn't.
They mean the same and are both correct Russian, but "тоже" is more common in colloquial speech while "также" is mostly used in written and formal speech.
Update: Oh, as a second thought, in this sentence, actually, "также" would shift the meaning of "too":
- Я ем яблоки. Она тоже ест яблоки. - I eat apples. She eats apples, too.
- Маша ест бананы, груши и апельсины. Она также ест яблоки. - Masha eats bananas, pears and oranges. She eats apples, too.
The best way that the difference between Анна and она could be explained to speakes of English is by looking at the difference between the "endless“ in “an END·less [АН·на] day“ and the "anew“ in “starting the day a·NEW [о·НА]“. The vowels are different but the tone and stress almost identical: on the capitalized sylable.