Free websites generally do not offer PHP (or offer it in a way that adds code to each page espousing the 'free' nature of the hoster) so won't run any fancy website stuff such as scripts -- you would be limited to what you can provide using HTML only.
Many of the weather software packages include free HTML-based templates that would work with a free hoster, if that hoster allows FTP to the website.
If you spend just a little bit a year, e-rice.net hosts many of the weather folks, offers PHP and is $10 to $20/year for hosting.. almost free. A 'regular' hoster will charge 8-12$/month for hosting and throw in a domain name .. that's only a couple of Lattes a month.
Best regards,
Ken