Difference between "I've added" and "I added"

Could anybody describe me difference between I've added and I added?

The key to the difference is "that has present consequences". If you are regarding the event as complete in the past, use "I added"; if you are considering its present relevance (eg the state of whatever you have added to) use "I have added". Note that there may not be any difference in the event: you can describe the same event either way, but you are focussing on different aspects of it. Generally you cannot use the perfect with a explicit time, unless that time is a period which includes the present:

I saw him yesterday
I saw him today
*I have seen him yesterday
I have seen him today

(The * means unacceptable)

To me, the difference between I didn't see him today and I haven't seen him today is that the first implies that the possibility of seeing him is over (perhaps he was only here this morning) whereas the second doesn't. I believe that North American usage is not the same in this regard, though.

I've added is present perfect, while I added is past tense.

The present perfect is a grammatical combination of the present tense and the perfect aspect, used to express a past event that has present consequences. An example is "I have eaten" (so I'm not hungry). Depending on the specific language, the events described by present perfects are not necessarily completed, as in "I have been eating" or "I have lived here for five years."

The present perfect is a compound tense in English, as in many other languages, meaning that it is formed by combining an auxiliary verb with the main verb. For example, in modern English, it is formed by combining a present-tense form of the auxiliary verb "to have" with the past participle of the main verb. In the above example, "have" is the auxiliary verb, whereas the past participle "eaten" is the main verb. The two verbs are sometimes labeled "V1" and "V2" in grammar instruction.

From Wikipedia

A past tense (abbreviated pst) is a grammatical tense that places an action or situation in the past of the current moment (in an absolute tense system), or prior to some other event, whether that is past, present, or future (in a relative tense system).1 Not all languages mark verbs for the past tense (Mandarin Chinese, for example, does not); in some languages, the grammatical expression of tense is mixed with the expression of mood and/or aspect (see Tense-aspect-mood).

Again from Wikipedia

One is a contraction for "I have added" while the other is "I added." The emphasis is in a slightly different place, depending on your usage. We would need a little more context to make any more meaningful comparison.

The format using I added refers to the question: "Who added some exercises to my homework?", and the answer "I did, I added some exercises to your homework." The format using I have added refers to the question: "What have you done to my homework?", to which the answer might be "I have added some exercises to your homework."