The first is marked 接
(short for 接続詞{せつぞくし} "conjunction"), while the second is divided into 接助
(short for 接続助詞{せつぞくじょし} "conjunctive particle") and 終助
(short for [終助詞]{しゅうじょし} "sentence-final particle").
So what's the difference? As it says, and as you can see from the examples, they are different parts of speech. The conjunction can appear at the beginning of a sentence, as in the example it gives:
「彼は頑固だ。けれども、話はわかる人間だ」
The conjunctive particle, however, appears after the 終止形{しゅうしけい} ("terminal form") of an inflectable word (such as a verb or adjective). The dictionary gives this example:
「年はとっているけれども、実に活動的だ」
Compare the examples. The former is two sentences long because the conjunction けれども
begins a new sentence; the latter is only one sentence long because the conjunctive particle けれども
attaches directly to the verb.
Finally, there's also the sentence-final particle けれども
. Much like the conjunctive particle けれども
, the sentence-final version appears after the terminal form of an inflectable word. Unlike the conjunctive particle, however, it ends the sentence. See this example from プログレッシブ和英中辞典:
お願いがあるんですけれど
I have a favor to ask of you.
In this answer, I've focused only on the difference between parts of speech. However, the sentence-final version of けれども
is also semantically different. Since that semantic difference has already been covered in a separate question, I will refer to it here—please see けど at the end of the sentence?.