There are some situations in which an employee can leave their job without giving any notice – but simply wanting to take up a new position elsewhere isn't one of them. Leaving without giving due notice will put you in breach of contract.
But if you are going to work for a rival, your present employer could apply to a court for an injunction to stop you working there, at least for the duration of your notice period (and possibly longer if there are appropriate restrictions in your employment contract).
There is such a thing as a “permanent record. If your former company gets purchased by a big company, they transfer your records to the new company. That means that your poor exit from a tiny company you never want to work for ever again, prevents you from ever being employed by the big multinational company that bought out your little company, even if all they did was fire everyone who worked for small company.