With milliseconds, the resolution of the timer isn't large enough. In most cases the callback won't be called more often than roughly 50 to 250 times per second, even when you set the interval to 1ms. See Timer resolution in browsers (as referred to by Sani Huttunen) for an explanation.
With 1000ms it will work better. But still the timer won't be fired when the tab is inactive, and may be delayed when the cpu is busy or another script is running on your page.
One solution is to not increment a counter, but to test how much time has actually passed since the previous call of the timer. That way, the timing remains accurate, even when the intervals have been delayed or paused inbetween.
This snippet will remember the start date, and on each timer interval, update seconds and milliseconds to the difference between the current time and the start time.
var start = new Date();
var milliseconds = 0;
var seconds = 0;
setInterval(function()
{
var now = new Date();
milliseconds = now.getTime() - start.getTime();
seconds = round(milliseconds / 1000);
}, 1000);
I've set the interval to 1000 again. You might set it shorter, but it will cost more performance.
Related question: How can I make setInterval also work when a tab is inactive in Chrome?