The deal with the EU centres more on the concept of 'a level playing field' than it does on 'free trade'.
i.e. The UK has agreed that the import duties it charges on good won't be massively higher than the EU charges on similar goods which we export to them, and vice versa. Similarly, we've largely agreed that our government won't provide massive subsidies to UK firms, enabling them to undercut rival firms within the EU, and the EU has likewise agreed that our firms won't face unfair pressure and companies within the EU that have been heavily subsidised themselves.
The EU's rules prevent it from removing all trade barriers with an outside country unless that country joins the European Economic Area, membership of which comes with a requirement to follow many of the EU's rules, or otherwise signs up to a deal to follow the main rules relating to trade. Norway, Iceland and Liechtenstein have all joined the EEA, whereas Switzerland has opted for a slightly looser agreement.
The UK was offered the chance to join the EEA, or to negotiate a 'Swiss-style' deal, but turned down those offers.
So now, for most purposes, trade between the UK and the EU is basically conducted upon much the same lines as that between the UK and China or the UK and the USA. That means, for example, if you send a birthday gift to a relative in France or Spain, you'll need to remember to attach a CN22 customs declaration to it, otherwise it won't get there. Further, unless it's a low-value gift, the recipient will probably have to pay charges to receive it.