How do hashtags on Instagram compare to hashtags on other social networks, like Twitter?

[Andrew]

If used correctly using hashtags in Instagram simply defines what the image is of or about, and can help bring together users with a similar interest or with similar images.

Hashtags on Twitter, among other things signify the topic of the tweet. Hashtags on Twitter, unlike Instagram start to 'Trend' when so many people are talking about them. This usually happens when people are discussing current affairs/events around the world/country.

Hashtags on Instagram are completely unrelated to hashtags on Twitter.

So yes, hashtags are/can be used in advertising, organizing, finding people with similar interests, and promotions. Promotions are especially used on Twitter, that's why most of the people who post them ask people to retweet them, - to get them trending, i.e.: noticed.