display:none means that the tag in question will not appear on the page at all (although you can still interact with it through the dom). There will be no space allocated for it between the other tags.

visibility:hidden means that unlike display:none, the tag is not visible, but space is allocated for it on the page. The tag is rendered, it just isn't seen on the page.

For example:

test | <span style="[style-tag-value]">Appropriate style in this tag</span> | test

Replacing [style-tag-value] with display:none results in:

test |   | test

Replacing [style-tag-value] with visibility:hidden results in:

test |                        | test

Basics

For controlling "cellpadding" in CSS, you can simply use padding on table cells. E.g. for 10px of "cellpadding":

td { 
    padding: 10px;
}

For "cellspacing", you can apply the border-spacing CSS property to your table. E.g. for 10px of "cellspacing":

table { 
    border-spacing: 10px;
    border-collapse: separate;
}

This property will even allow separate horizontal and vertical spacing, something you couldn't do with old-school "cellspacing".

Issues in IE ≤ 7

This will work in almost all popular browsers except for Internet Explorer up through Internet Explorer 7, where you're almost out of luck. I say "almost" because these browsers still support the border-collapse property, which merges the borders of adjoining table cells. If you're trying to eliminate cellspacing (that is, cellspacing="0") then border-collapse:collapse should have the same effect: no space between table cells. This support is buggy, though, as it does not override an existing cellspacing HTML attribute on the table element.

In short: for non-Internet Explorer 5-7 browsers, border-spacing handles you. For Internet Explorer, if your situation is just right (you want 0 cellspacing and your table doesn't have it defined already), you can use border-collapse:collapse.

table { 
    border-spacing: 0;
    border-collapse: collapse;
}

Note: For a great overview of CSS properties that one can apply to tables and for which browsers, see this fantastic Quirksmode page.