Three different situations hide behind fill a PDF form, and telling them apart explains both the cannot type and the saves blank problems:

The interactive form, the easy case: a PDF with real form fields, clickable boxes that accept typing, fills in any competent PDF reader including the free Adobe Acrobat Reader and File, Save writes the filled version permanently. When this saves blank, the cause is almost always filling it in a browser's built in PDF viewer which does not always save field data, the fix being to download the PDF and open it in an actual PDF reader before filling, then save.

The flat form, the cannot type case: a PDF that looks like a form but has no interactive fields, just lines and boxes drawn on the page, gives nowhere to click because there is nothing to click. The fix is a reader with a Fill and Sign or typewriter feature, Acrobat Reader's Fill and Sign lets you place text anywhere on the page regardless of fields, typing over the flat form, then save. This is the tool for the form that will not let you type normally.

The saves blank on reopen, beyond the browser cause: some forms are set by their creator to not allow saving filled data, a form permission, showing filled on screen but blank when reopened by design. Fill and Sign text often survives where field data does not, and printing to a new PDF, print to PDF of the filled form, flattens your entries permanently into a fresh file that always reopens filled, the reliable escape from a form that refuses to save its own fields.

The reliable habit for all of it: download rather than fill in browser, open in Acrobat Reader, use form fields if present or Fill and Sign if not, save and for anything important print to a new PDF as a flattened permanent copy, the version that cannot lose its data because the data is now part of the page rather than a fillable field.