(plain form verb/i adjective/na adjective+な)+ものがある (Kanji 物がある, but I think it's usually written in Hiragana), can mean:
and is used in regards with things the speaker felt, for expressing feelings while describing the characteristics of something, or that a certain characteristic can be seen, for example:
"The cat which disappeared half a year ago has returned. To me it's special, and I feel very happy."
"This sentence has still got some way to go, but it twinkles at every turn."
"This painting feels like it pulls a person into it."
You can also use 見られる, 認められる etc in place of ある here.
(Sources: 日本語文型辞典, どんな時どう使う日本語文型辞典500, the Kanzen master JLPT 2 grammar book).
It can also mean "there is a thing (generally a tangible thing)" (with the Kanji 物がある).
ものがある can also mean "there is a person" (written with the Kanji 者がある)
It can mean "there are times when..." when expressed as plain form verb+ことがある and "there are times when it doesn't" when expressed as plain negative verb+ことがある
It can express that something has been experienced when expressed as plain past+ことがある, and that something hasn't been experienced when expressed as plain past+ことがない.
It can also mean "there is a thing (generally abstract)" as well. (It is structurally ambiguous between an appositive clause and a relative clause)
The question you linked to describes these usages in more detail, so I think I might leave those for now.
There are probably many other different usages too, but the two are generally quite different to each other as far as I know. I think as a (possibly broad generalization) that ものがある tends to be about feelings and tangible things and ことがある tends to be about abstract things, experiences and occurrences etc.