There may be many reasons.
The capital gains tax regime already distinguishes between homes for investment purposes and homes for living in. People do not pay capital gains tax on the home they live in, but if they do not live in it, they do pay. The reason that people buying homes are overtaxed has been laid out by many colleagues, so I will not overlay that, but clearly we need labour mobility and for people to be able to live in appropriate housing. If we overtax housing, we end up with lots of people living in inappropriate houses, not least the home hoarders at the end of their working lives and the empty nesters whose kids have gone and who live in too big a house and do not want to move. That has a damaging impact on labour mobility, as people want to move around the country. It is the most economically damaging tax.
The reason I say that people buying second homes for whatever purpose or homes for investment are undertaxed is that stamp duty is a transaction tax. All other transaction taxes—we can argue about whether VAT or excise duty on petrol are transaction taxes—are flat-rated. People pay the same rate, whatever the value. We pay 20% VAT whether we are buying something for £10, £100 or £100,000. Stamp duty, however, is a progressive transaction tax, with a lower rate at the bottom end that progressively goes up, as it is designed now. That zero rate and lower rate for lower value properties is for social reasons. It is basically to help first-time buyers get on the property ladder, which is welcome, and to help people get up the property ladder. That is a valid social reason to have progressive stamp duty, but that reason does not apply to people buying second homes or investment properties. There is no reason to have a graduated stamp duty to help property investors get on to the property investment merry-go-round.