There are actually quite a few differences.  

The biggest difference is that Bitdice (BD) allows investment and PD does not.

The biggest difference for low rollers is that there is no faucet.  So the chat is less active on BD and much less beggary.  In fact, there is almost no begging on BD as there is just no tolerance for it and the site is coded so that you can't chat if you use TOR, so when you are banned, you are GONE.  I really like it.

PD's tipbot fires randomly on individuals based on chat activity.  BD's tipbot fires randomly once per hour or so on everyone who chats during its window - which is also random.

There is no minimum tip (? or it's pretty low) on BD like there is on PD.  This is possible since there is no faucet.  BD has an affiliated coin exchange, simpleswap, as well as is part of a family of sites - litedice, reddice, and dogedice.  The chats are linked though the sites are distinct.  I like altcoins and so that is a nice feature.

So IMO there are a lot of differences and I've started to spend a lot more time on BD compared to my old favorite PD.

Good Luck!