Other than the high cost, lithium-ion batteries are superior to other technologies in just about every way. They are lighter, can be discharged repeated to almost 0% without damage, and using the right equipment can be charged much faster. However, until the technology becomes more mature, you need to learn a lot about these batteries before switching over.
Most Li-ion batteries intended for marine use are designed for deep cycle house battery or trolling motor use. But we are now seeing lithium batteries designed for starting and dual-purpose applications. For a marine starting battery, you need to find one with a 1000 CCA rating. The deep cycle versions won't be able to deliver that amount of instant current. But there are a number of nice dual-purpose Li-ion batteries that will do a good job both for starting and house purposes. But if you are going to use a bank to power a trolling motor, go with deep-cycle.
You can charge Li-ion batteries directly from an outboard alternator, but this is not ideal. Without going into a lot of technical stuff, the life span of both the battery and the alternator will be reduced. You need a different battery management system to optimally charge Li-ion batteries and to protect the alternator. Several Li-ion starting batteries that are designed as drop-in replacements for flooded cell or AGM batteries have built-in battery management systems. Unfortunately, not all of these work very well. So it is recommended to spend an extra $300-$500 for an external alternator regulator designed specifically for use with Li-ion cells like the Wakespeed WS500 or Balmar MC-614-H. These allow the batteries to be charged rapidly and protect the alternator. Another solution for people with a bank of L-ion batteries used with a trolling motor is to install a DC to DC charger. Here you run the alternator to your conventional batteries and connect one of your flooded cell or AGM batteries to the charger. Once the conventional battery is fully charged, something like 10 amps is used to charge the Li-ion bank. This charges the Li-ion bank a lot slower than a dedicated regulator.
Li-ion chemistries are very safe now, so that shouldn't be an issue. And new chemistries like lithium iron phosphate allows Li-ion batteries to be charged at temperatures below freezing, which is still an issue with earlier chemistries.