What are the major pros and cons of using coin's own/designated wallet?
That varies quite a bit based on which coin you're talking about. For example, with Bitcoin, using the core/QT client has the infamously long syncing process for new wallets, and the storage requirement is high and ever-increasing. (Although pruning is an option.)
Many other "default" wallets are full node wallets that need to sync the entire blockchain. For example, Litecoin, although the blockchain size is much smaller than that of Bitcoin.
However, if you trust the coin enough to invest directly or invest mining time, then it seems that you would trust the development team behind the default client. (Assuming you download it from the correct source and trust the signature.) Using web-based wallets or other 3rd party wallet clients requires you to trust a separate development team. (Although many 3rd party wallets are good, just wait for them to be verified independently.) Coins such as Ripple that do not have a core wallet client are a different story...