Travel insurance bundled together when you're being sold a holiday typically isn't great value.
The most obvious things to compare are the levels of cover, the excess (€ of any claim you have to pay yourself) and any exclusions
I haven't looked got details of the Budget policy but €150 for a single trip (probably within the EU?) seems very high indeed ... When I go to budgettravel.ie it quotes me €115 for an annual multi trip policy.
The insure.ie policy looks pretty comprehensive. ... only thing is the excess is reasonably high €75/€100(medical) ... Also the coverage is maybe a bit stingy on duration and trip length compared to other annual policies -it covers you for 90 days of the whole year for any number of trips (up to 1 month long)
Even if you typically only tend to go on one holiday a year you could still get two holidays worth of cover out of it provided in the second year you took your holiday at least two weeks earlier
It used to be the case some years ago that package holiday providers could force you to take out "their" insurance (which is really nothing to do with them anyway just a branded product from some big insurance group) .. but this is no longer the case as far as I'm aware. I believe they can insist that you have insurance and request evidence of it (travel insurance is sensible of course), but they cannot insist that you have their insurance.