Very much depends on the tone, the suitability of the event and how it came to be asked.
For instance if it came up openly in conversation, I was given full details and they said 'you'd be very welcome' to something that was quite open in terms of numbers or who was invited (so, not Christmas day), I would take it as them wanting me there, just me not being part of the initial group in mind, which is fine.
If it was more that I overheard or joined a conversation between people already going, and the invitation was given offhand, I might assume they were asking out of politeness. That wouldn't necessarily mean I wasn't welcome, but i would look at the type of event and think whether one more would make a difference either way before actually going.
If it's something like a house party or informal BBQ then it's quite normal for people to be invited ad hoc who may not have been on the original list. A gathering of close friends who live miles apart, family party etc, not so much.
Also, if you don't know someone well or aren't very self assured, it can feel a bit like you're imposing by inviting them. This offhand 'you're welcome if you like' means job done but you've not put any expectations on them. But not ideal if they're then wondering if you meant it.
'Come if you want' is perfunctory though and I wouldn't feel particularly welcome.