Greetings

, without creating animosity, or anything of the like, I have a question regarding retirement plans I would like to get real life experiences from any and all that might have them to share.
The take home packet I received from my new employer has some things different than the last. The new company is a Not-For-Profit (NFP) organization and therefore has different packages.
403b, I have a 401k and understand how it works, but have x days remaining to close it out. 403b reads similar, but different, what is the real difference?
Can one transfer 401k funds to 403b funds sans penalty? Different institutions as well. Current Fidelity, going to Vanguard.
What options might there be, as this is an area I'm clueless in.
Bottom line, I must close out my 401K from my previous employers corporate account within 90 days.