What is the difference in meaning between להחמיץ and לפספס?
Is there any difference meaning between the two verbs להחמיץ and לפספס? If so, when does use each?
To miss is basically "לפספס", you can use it for pretty much everything; "להחמיץ" is more formal and used in specific sentences like "to miss an event" - "להחמיץ אירוע" et cetera.
One rarely uses "להחמיץ" since it's used in letters or when speaking formally.
I'd give you a list of sentences but I just can't think of any right now.
Basically the same thing, except לפספס is slang. This is actually not intuitive to Hebrew speakers because לפספס is used a lot more, while להחמיץ is used in very specific cases, like when missing leaves you with a feeling of missing out (although even then you often use לפספס).
However, in a way, לפספס isn't a "real word" in Hebrew. It is marked as slang by Even Shoshan, the leading Hebrew dictionary, but the real proof is that it doesn't follow standard Hebrew construction rules. With those rules, פִּספֵּס would be written and pronounced pispes, and לפספס would be lefaspes. This is not the case however.
Le'faas'fes, and no one cannot say "אני מפספס אותך" it is wrong, we have a special verb for that, "להתגעגע" - "אני מתגעגע אלייך"
You can say אני מפספס אותך, but it has a different meaning. It's like in: "I miss you every time, when I arrive, you are already gone" (אני מפספס אותך כל הזמן, כשאני מגיע, אתה כבר לא נמצא.). When you want to say: "I miss you, I didn't see you for five years", you can't use the verb לפספס, you should use the verb להתגעגע. For example, אני מתגעגע אליך, לא ראיתי אותך כבר חמש שנים.