Today in the morning at 10am I received a message regarding a schedule change of my flight from Krabi to Bangkok today: Instead of arriving at 10:45, the flight is now scheduled to arrive at 02:10. (Vietjet Airlines). My connecting flight is with another company (EVA airlines) departing to Vienna at 02:45 from Bangkok. Unfortunately, this change in time makes it impossible to reach my flight back to the EU. I have now booked a new ticket which is way more expensive to reach Bangkok airport in time from Krabi. Vietjet claims that the only thing they can do is refund my original ticket but they won't pay for my new flight. I have not accepted the refund yet as the form said I would give away any other right towards compensation. Is there something I can do more except refunding my original ticket?
This is unfortunately the risk you take whenever you book "connecting" flights on separate tickets - the risk of the first flight being delayed is on you. Whilst leaving 4 hours between flights as you've done can potentially reduce this risk, clearly this example shows that's not always the case.
Vietjet airlines is correct that they have no responsibility for your second flight, or for paying for an equivalent flight if you choose not to fly with them.
There are three potential courses of action you have when something like this occurs :
Contact the second airline (EVA) and explain the situation. They may choose to take pity on you and allow you to change to a later flight at no cost. The odds of them doing this are low, but you never know if you don't try...
Claim whatever compensation the airline is required to provide under local laws. As far as I'm aware, Thailand has no such compensation laws so that's out in this case.
Claim the additional costs under your travel insurance. (You do have travel insurance, right?) Even if you don't have explicitly purchased travel insurance (you should!), if you purchased your tickets on a credit card then you may find you are covered by the free insurance from there.
Note that not all travel insurance policies are created equal. Some may cover your additional costs as a result of this delay, whilst others might simply refund the cost of the missed/delay flight, but still leave you liable for a replacement flight which will likely be much more expensive. Be sure to check your policy or talk with the insurance company before presuming what will be covered.