Just to supplement jknappen's excellent answer, we don't see a lot of inherited vocabulary shared between Italic and Celtic, simply due to the timescales involved. When there are cognates between the two, they usually come all the way from PIE, or were loaned at a later point.
The main evidence for Proto-Italo-Celtic comes from certain grammatical features common to Proto-Italic and Proto-Celtic, but not found in other Indo-European languages.
For example, PIE didn't really have tense distinctions, so when its daughter languages created a past tense, they did so in different ways. Both Italic and Celtic—and no others that I'm aware of—took two particular unrelated forms (the PIE perfect and aorist) and gave them the same meaning. So some verbs form their past tense with reduplication of the first consonant (the PIE perfect), while others use a suffix -s- or -t-, or a lengthened vowel (the PIE aorist). There are also some innovations common to the two families, such as a vowel -ā- marking the subjunctive (compare Latin ferat and Old Irish bera, "he/she would carry", versus Ancient Greek phérēi).
The question remains whether there was actually ever a "Proto-Italo-Celtic" language, or whether these features spread through language contact, or are sheer coincidence. To my knowledge, the jury's still out on that: there's much less evidence for this than for, say, Proto-Germanic, which is uncontroversial.
