Our family went for 5 days in March 2011 and 4 days in March 2012. The weather was pretty stable - obviously not exactly hot, but we only got minor rain for 20 minutes or so every now and then. Everything was running as normal, nothing stops due to weather etc.
You'll get the normal afternoon parade in Disneyland Park (no night parade), Stars and Cars in the Studios Park, and Disney Dreams in the evening. Dreams has minor firework elements, but you won't get a full firework display unless you are there for the Welsh St David's Festival or Irish St Patrick's Weekend. If you've got lucky and accidentally booked during one of these times, there will be extra entertainment and shows.
If you are used to going in summer holidays or peak times (Halloween/Christmas) you may notice slightly more ride closures for refurbs than you are used to. Expect at least 2 attractions to be closed for the duration of your stay. In March/April Pirates of the Caribbean seems to be refurbished a lot. However on the plus side, Monday-Thursday you'll find the parks aren't too busy at all and there will be little to no queuing!

[EDIT] I should point out that its not 100% confirmed that Dreams will still be running in 2015 (its only confirmed til Jan 1st) but I can't see it stopping any time soon
