Both shopping cart price rules and catalog price rules will apply to the product prices as they are used in the cart. The difference between the two is a shopping cart price rule is a discount that only shows once the item is in the cart, typically based on criteria which the line items in the cart meet. Catalog price rules are changes to the products in the catalog which are displayed to the customer when browsing the site, and the prices seen should indeed be reflected by the cart when the item is added to the cart.
For catalog price rules to work, however, the Magento cron job must be configured to run on the server running Magento. The reason is, catalog price rules are computationally expensive, so they are only calculated when the
catalogrule_apply_all crontab is run and when the rule is saved and applied via the Magento admin.
The indexes created when applying the rules are date specific and typically calculated one or two days in advance. So if the cron is not setup and you've saved and applied the rule in the admin, they will show properly for a time, but then suddenly cease working.