Difference between an "activist" hedge fund and a PE firm

One point, but not necessarily the most important difference, would be the structure.

PE firms tend to do a large capital raise, close the fund, then go about their business and return capital after a long period of time, 5, 10, 15 years etc. HFs tend to be more liquid, for example some activist funds are even monthly/monthly for redemptions and subscriptions of new capital.

I don't want to make too many comments on the difference in strategy since the activist funds I have spoken to have been much smaller, around ~250mln, but being a long lock vehicle I would assume PE funds do not need to worry too much about the mark-to-market prices, so they can focus much more on long-term value creation. Activist funds need to report prices usually monthly so that investors can decide whether to redeem.

PE firms will usually be buying out entire businesses with the intention of selling it on in a few years time, whereas an activist hedge fund could simply be taking a stake large enough that they can exert a little bit of influence on the business, so a stake just big enough to get them a seat on the board perhaps, yet small enough that it remains liquid.

Another point worth making is that a lot of hedge funds will not touch private equity, which also means they cant buy out the entire business, since it can become a problem when reporting fund NAV. Those I can think of that will do private equity will usually put it in a side pocket and have a 3rd party value it for them. Most of these HFs will have a limit to the amount of side-pocket exposure they will give investors which limits what they can do