You can potentially magnify whatever gain you have investing if the Krona becomes weaker, or the Euro becomes stronger. If you have a 50% gain and the currency gains 10% as well you really have a 60%+ gain unrealized (before conversion fees etc). This goes equally for dividends.
Over the long term currency between two established trading partners like the Swedes and Euro zone is relatively cyclical. If one is outside of the historical strength/weakness. You can play that to your advantage when you invest.
Access to different companies investment options than your local choices. Diversification can be great.
The exact opposite of all the pros mentioned above. If the Krona becomes stronger you will lose out money if you want to sell at a bad time.
Strength of a currency against another is relatively unpredictable, and in the short term this isn't a great thing. If you are forced to get out of a position you may end up losing money on your transaction depending on how the currency is doing.
Currency transactions are typically expensive. My experience is something like 2-5% (both ways!!!). This eats away at dividends, and capital gains over time. So unless you expect a big gain, it usually isn't worth the trouble. Remember you have a fee for switching from Kron to Euro, and then back from Euro to Kron.
Adds complexity to your investments, because you have to consider what you think will happen to the stock, etc. and what will happen to the currency. It is worth looking into different options to hedge the currency part of your transactions if you desire stability. However they might not be accessible to retail investors, or be practical for them due to cost.