It's hard to have a light Monero client, because it has to have some data indexed about all past transactions, whereas a Bitcoin client only needs to maintain the MUCH smaller UTXO set, i.e. the unspent transaction outputs. This is one of the ways in which Monero scales poorly.

Another problem is that Monero's PoW is expensive to verify where a good PoW should be instantly verifiable. In fact, it's considered so expensive that by default, the Monero client uses a checkpoint block without verifying all the PoW that led to the block.