The difference "了" or "过" is, as Wozitoya said, “了“is "the perfective aspect", “过“ is "the experiential perfective aspect". And “了“ has two different uses in grammar. 1. 我吃了。 Then “了“ is the aspectual particle. As you noted, used after the verb to indicate past tense. 2.我吃过了。 This “了“ is the modal particles(语气助词). used to express mood, or an expression of how a statement relates to reality and/or intent. This goes to end of the sentence. This is my understanding. Sorry,I'm not a native,I can't describe well.

"过" refers to something that has been done. "了" added to the end of the sentence that change the statement to past tense. 我吃过了, here "过" links to the verb "吃", it became a past tense . 我吃过 我吃过喇 ! The above sentenses are past tense. Chinese used to add "了" to the end of the sentense in forming a past tense. So, adding it to the end of the sentence are also considered to be correct, In your example, "了" can be omitted.

There's a small difference between the two. Verbs in Chinese remain in the present tense, (看, see,吃，eat, 喝drink）unless you add something before or after it. 过 placed after a verb changes the meaning to past tense. Example: 吃 = eat, 吃过 = "has eaten" (in the past). 你吃过月饼吗- have you eaten moon cake? 你看过这部电影吗- have you seen this movie? 你去过上海吗- have you been to Shang Hai? 了, when placed after a verb, just expresses that the action has been completed. 吃 = eat 吃了 = ate Example: 我吃了葡萄和草莓。- I ate grapes and strawberries. Another example: the verb 买, means "to buy" when 了 is placed after it, it becomes 买了 "bought" 我买了一本杂志- I bought one magazine. In Chinese, people sometimes use them the same way but 过 tells of experience, while 了 just indicates the action being completed. Asking 你吃了饺子吗 and 你吃过饺子吗 expresses two different things. 你吃了饺子吗 is asking in general "did you eat dumplings" the person can be referring to today or yesterday or this morning. 你吃过饺子吗- is really asking "have you ever had dumplings?" meaning have you had it in the past?

“过”助动词，代表已经发生过，了是语气助词，可省略，学习语言应多听多说，很多时候是习惯用语，没有为什么，说得多了就自然说出来了，也不必太费力去研究区别，除非你是汉语学家，若只为交流则得不偿失

这个问题不是三言两语可以解释清楚的，因为“了”字是外国学生最难掌握的，已经研究很久很久了，我手上关于这两个词的论文都不懂有多少篇了。我建议你去看看一些专业的语法书，还有就是注意平时的使用，错了就记下来。