Where do I start? You are doing the right thing by getting hold of samples of the language and listening to them, but here are a few pointers. I think some of them have already been mentioned here but it doesn't hurt to say them again!
The main difference is that the stress usually falls on the first syllable in Ulster, and there is a tendency for vowels which aren't in the first syllable to become that blurry indistinct mid-vowel which linguists call a schwa. Thus the word for salmon, bradán, is pronounced BRADDun in Ulster. In southern dialects where the stress is on another syllable, it's often something like burDAWN. The word for bottle, buidéal, is usually BWIDJul to us, but bwidALE to a southerner.
The vowels are generally different. An o will frequently sound more like an a. Thus ina chodladh (asleep) would be ina holla in most dialects but almost ina halla in Donegal. The á is a very different sound in Donegal. In the south it's an aw, but in the north it's somewhere between an ah and an eh. The ú of words like tú or lú is very rounded in Donegal - almost the way a French speaker does a u.
The main difference with consonants is probably the way a bh and an mh are often pronounced as w rather than v. In the south they're nearly always v. In Ulster Irish, it's v when slender (that is, next to an i or an e), so these are all v sounds: bhean, mhír, bhí, mhíolta, bheannaithe, bheannaigh, mhill, mheall, bhinn, bhéal. It's also a v when there is an r or an l or an n next to a bh or mh, as in: bhris, bhréag, bhrí, bhriathar, mhná. The rest of the time, in Ulster, these are like a w. So bhuí, bhog, bhfuair, bhain, mhór, mharaigh, mharbh, mhaol, mharcáil are all with a w sound.
Another difference is the slender d of déanamh, díreach, dian. This is more like an English j than in the other dialects, though not exactly the same. And the slender r is sometimes pronounced like a y and sometimes a bit like a z. It's a strange sound. See if you can pick any out from T na G!