The difference is in the Schwa, which is a lazy sound native English speakers use. It can vary a little in different parts of the world, but is generally considered to sound like "uh" or "ih". In this example, I have a British or New Zealand accent, and I would pronounce inter as in-tur (the 'r' is soft), and alter as ul-tuh (the 'r' is silent).

The important difference here is the stress patterns of the two words. It has nothing to do with whether or not your accent is rhotic (pronouncing the 'r' after a vowel). The verb 'infer' has the stress on the second syllable. The 'fer' part of the word is stressed, so this is a long vowel - the same as in the word 'fur'. As you can see from the phonemic transcription in your question, it doesn't matter whether you pronounce the 'r' after it or not. The vowel is the same. In the word 'alter', the stress is on the first syllable. This means that the 'er' sound is unstressed. All unstressed syllables in English are either a short /i/ or a weak 'schwa' sound, like an 'uh' but with no effort or force behind it. And once again, you whether or not you pronounce an 'r' after it makes no difference - it's still an unstressed syllable with a schwa. Understanding stress is crucial to good pronunciation. Note that your dictionary will tell you where the stress lies by marking the stressed syllable with a ' mark. For example: in'fer as opposed to 'alter.

Su.Ki. writes a very good explanation, though she has used the example of the verb "infer", and in her example I agree 100% with her. However, inter is a curious case. The word could be a verb (to bury someone) in which case it is pronounce the same way as "infer", or it could be a prefix, and then the situation gets a little tricky. When we use inter as a prefix we usually stress the first syllable and not the second syllable, or we don't stress either syllable instead preferring to stress one of the syllables of the word after the prefix.

Brian is correct: Americans (except in Boston!) pronounce the "r" sound much more than the British do. So they would sound exactly the same in most parts of the USA. A British accent would drop the "r" at the end of "alter," resulting in something like "all-tuh."

Thank you Brian :)