Can two people with and without Reaper of Souls play together, and will they cap at 60 together?

I have a friend I'd like to play with who has Diablo 3 with the Reaper of Souls expansion. I don't have the expansion myself.

Will my friend and I be able to play together?

Further, when we hit level 60, I'm pretty sure I won't be able to keep levelling up - but will they keep going on past level 60? (That would be problematic, because of the power level difference!)

Will there be any significant differences in our experience that might be disruptive to play due to only one of us having the expansion?

You will be able to play together under certain circumstances. Essentially, you must make the game, and in that situation, you will always be able to play together. While in the same game, there will be no difference in the experience of the play.

You will not be able to go to Act V nor will you be able to join in adventure mode.

If you make a game where you are 60 and your friend is 60, you will be capped. Your friend will not, and will continue leveling. As your friend levels, the monsters and items will become higher than level 60. This will mean you end up salvaging into expansion account bound materials for crafting, however you will not be able to use them because you are still level 60 and your blacksmith is capped at level 10.

So, yes you can play together, but really, there is so much more to the expansion you will have much more fun just giving in and getting it.

Yes you can still play together. As stated by this blue post on the forum :

If you have Reaper of Souls, then you can join in games that have been created by players that do not have the expansion. However, players that don’t have the expansion won’t be able to join games that were created by accounts that have been upgraded to Reaper of Souls.

However (as its not precised) If you own Reaper of Souls and create a game that does not require the expansion (below level 60 and not at act V) people without Reaper of Souls will be able to join you.

Your friend won't cap at 60 and will keep on leveling up until 70.