I've read what Kraken have mentioned in the difference guide but I haven't understood it well.
I plan to mine ethereum. What fits better for me?
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Briefly, ICAP addresses are safer to use, since they have a checksum which provides protection against mistyping an address. But ICAP adoption isn't as widespread currently. For mining you can use either, since a HEX address, when needed, can always be obtained from an ICAP address.
ICAP is the best option for wallets that support it.
So the direct answer to your question is a list of clients which support it:
ethseems to support ICAP but it's integration looks experimental. I would use hex instead.
gethdoes not support ICAP. Use HEX instead. Basic ICAP support is planned for future releases, maybe starting at version 1.4.x.
mistdoes not support ICAP. Use HEX instead. Source.
TL;DR use HEX for now.