If you're playing a solo franchise, there is really only one reason to play a cloud franchise and that is if you want to take advantage of the feature to start in a particular week of the season (with the trades and draft picks exchanged). Cloud franchises do advance more quickly from week-to-week, but it's a difference of a few seconds (many other games have far longer load times, so I hardly think that's a big deal). Other people will tout that they get franchise updates (server-based) more quickly - the reality is that, to my knowledge, EA hasn't done any cloud-based tuning/fixes at all this year (and I don't believe they did last year either).
Aside from the obvious that you may encounter server issues, the major downside of a cloud franchise is being able to take control of CPU teams mid-franchise. I
believe you cannot do so without having a separate login ID. Sometimes, you have to do some roster 'cleanup' for the CPU as they don't always handle injuries well. Staying offline gives you the flexibility to temporarily take control of a CPU team to make tweaks for them (or, if you just want to do something to create a fun storyline for your franchise).
I have not tested online franchise from a gameplay perspective but some folks I trust have commented that there is little-to-no difference.