If you're saving for retirement in a 401(k) or picking mutual funds, most advisors will tell you to spread out over different types of funds. This means that while you should pick some higher risk funds because of the greater return, you should also balance those with some "moderate growth" (safer bets) and the safest funds that offer little return, but little risk as well. But let's say the government agreed to completely indemnify you for all investing losses, why would you pick those safer stocks? You'd go all in on the riskier investments.
So, you have banks that are too big to fail. They know this and thus are more free to make risky investments. The government actually encourages this behavior. Why? Risky investments generate economic activity, economic activity will partialy inflate GDP. Inflated GDP leads to more consumption, consumption begets further economic growth and conceivably this cycle could get us out of this particular recession. The problem is that once again, we've done it on the backs of a bubble and when the next bubble pops it may be more painful than this one.
Should we let vital businesses fail?
This question assumes that any one business entity is vital. Failure a business entity in a free market economy is NOT the end of the world. It's an opportunity. When a business fails in a market where there is still demand, innovative new companies WILL come to replace the old. Additionally, it's not as if the failed banks assets dissipate into nothingness, they are liquidated and carried forward by a successor company.
What happens when the government bails them out instead is that money is taken from a productive segment of the economy and used to prop up a failing segment. Keeping in mind that the segment is most likely failing because of a lack of innovation or market adaptation. Those practices, instead of being eliminated with the failure of a business or businesses are instead VALIDATED. In other words, instead of companies adapting and evolving to do better, they're encouraged to continue the same risky behavior that caused the most recent collapse.