Caticlan is a a much better airport to fly into, because it's just a 15-minute ferry ride away from the Boracay island, whereas Kalibo is a 2-hour bus + ferry ride away. However, Caticlan airport is also much smaller and route is served by propeller planes, so from time to time they get cancelled or diverted to Kalibo
Caticlan airport's official name is Godofredo P. Ramos Airport but we all call it Caticlan airport. Additionally, it's sometimes also called "Boracay airport". So if you hear any of these names, you know it's referring to the Caticlan airport.
Flight time is 1 hour to both of these airports from Manila
Prices are similar to both of these airport. One-way fares is around $50-60 USD, and round trip tickets are $90-$100 USD. Cebu Pacific, Philippine Airlines, Air Asia, Cebgo all have non stop flights
From Caticlan Airport to Boracay
Once you get to Caticlan airport, you need to take a van or motor taxi (called tricycles) to the Caticlan Jetty, which is where the ferry to Boracay departs. This should cost you 50 pesos (about $1USD) on tricycle, and takes around 5-10 minutes
From Caticlan Jetty, you need to get on a ferry to Boracay. The ferry costs 225 PHP ($4USD) total, made up of 50 PHP in ferry fare, 75 PHP in environmental fees, and 100 PHP in port fees. These ferries run between 5am and 9pm everyday
The ferry takes 15-20 minutes to get to Boracay's Cagban Jetty. From there, take a tricycle or van to your hotel. Negotiate the fare, it should be no more than 100-200 PHP ($2-4 USD)
Alternatively, you can book a private transfer from the airport to your hotel on Boracay for around $10 USD. Use websites like Tripadvisor or Klook to find these transfers, there are lots of them on these sites! The benefit of using these transfers is that they are usually door-to-door, meaning they they take care of everything from the Caticlan airport to your hotel on Boracay, so you don't have to figure out the ferry part of your trip
From Kalibo Airport to Boracay
From Kalibo airport, my suggestion is to book transfers ahead of time through websites like Klook or Tripadvisor. The transfer operators will normally take care of the ferry tickets too so a single booking gets you from Kalibo airport all the way to your hotel, saving you the hassle of having to figure out ferries on your own. These transfers should cost around $12-$15USD.
If you don't want to book ahead of time, you can easily find ways to get to Boracay from Kalibo airport after you arrive. There will be lots of taxis, vans, and buses to take you to Caticlan Jetty. From Caticlan Jetty you can buy a ferry ticket to Boracay's Cagban Jetty for 225 PHP ($4USD). Once you get to Cagban Jetty after 15 minutes on the ferry, take a van or taxi to your hotel for 100-200 PHP ($2-4USD)
The total travel time from Kalibo airport to your hotel on Boracay is 2.5-4 hours (lower end of this scale if you book a transfer and higher end of this scale if you DIY)