Druid is my main on alliance (has been since 2004), shaman has become my main on horde. I love both classes and characters almost equally, though druid holds a special place for me since I’ve been playing it from the start.
Balance and elemental play very similarly in concept. They have similar builder/spender abilities and similar resource usage. Lightning bolt/Solar Wrath, Lava burst/lunar strike, earthshock/starsurge, earthquake/starfall, flame shock/moonfire, they all measure up pretty well.
Shaman suffers whenever fights dont last very long though. My druid starts at 50 astral power and can build it pretty quickly and easily with instant casts, but the shaman always starts from zero maelstrom and it takes a lot longer to build up to decent levels. Once you get rolling, shaman works really well, but it’s meant for longer fights and has a hard time on trash that dies quick.
Druid definitely has an edge in utility with better heals, roots, bird form, etc. Also has a built in slowfall for moonkin form with the flap ability. Shaman has some utility in tremor totem, earthbind totem, purge, etc. but they dont see use as often as the druid’s utilities do. Both have a ‘travel form’ (flight not withstanding) with ghost wolf / stag. Ghost wolf isnt as fast (stag goes normal ground mount speed, ghost wolf is only 30% / 50%-60% with talents), but ghost wolf can be used inside as well.
Druids have stealth, and shadowmeld as a vanish if you’re a night elf druid, and that helps a lot when soloing/questing whereas a shaman has to fight through everything. However, balance druid can - especially at lower gear levels - be extremely squishy and have a hard time with tough mobs/elites, whereas shaman has earth elemental to tank in a pinch.
Only way to really decide which one you like better is to play them. I play both, and love both.