In what could be called “soft” Arminianism (whether Protestant or Catholic), God persists in saving some sinners from sin, once the sinners have (in effect) convinced God to do so.
A Southern Baptist “soft” Arm would say that if you seriously confessed your sins to God and asked Jesus into your heart (or whatever), you’re safe, God will make sure you’re saved even if you fall back later. (Except they’d say a major fallback indicates you weren’t really serious to begin with. This introduces the usual experiential problem.)
A Roman Catholic “soft” Arm, which is the position they ended up with officially, would say that if you’re baptized by someone who is himself or herself a Christian (usually a priest although on this point I think they allow exceptions for special/emergency circumstances), God will persist in saving you. Thus the importance of infant baptism. You might have to go through a lot of purgatory in the eternal fire proportionate to a backslide (a factor the Southern Baptist typically would deny), but God will still bring you through.
A “hard” Arm (like a specially hardcore “freewill Baptist”) would say that there’s no way to guarantee God will save you before you die, other than to keep the faith perfectly. Or sufficiently perfectly. You know. 
A Calv (Protestant or Roman Catholic, like Augustine) by contrast would insist that no one has to convince God to save anyone; in fact people cannot convince God to save anyone. God already either intends to save you from sin or He doesn’t. If He does, you’ve been elected for salvation by God, and He’ll get it done (by purgatory if necessary, per Calvish RCs like Augustine) – various historical events (like baptism by believers in Catholicism) will be signs that God has elected someone for salvation (and so has allowed and/or brought about the events), but He intended to save whoever He saves from the beginning. (The evident signs may or may not be exclusive. RCs used to teach that unless someone was baptized in this life God wouldn’t save them; now they teach that God may save those post-mortem who weren’t baptized in this life, or not baptized by a proper representative. The baptism is directly related either way to God’s intention to persist; Arm and Calv RCs debate, or once debated, over what precisely the baptism indicated.)
Arms (like my mother) like to say they have just the same assurance of persistence as Calvs, but that isn’t exactly true: there’s a big and important technical distinction, if God has to be convinced to persist in saving someone or if that persistence is by His gracious and original choice.
The worry of whether God will persist for every individual, is in different ways a main factor in conversion from Arm to Calv or from Calv to Arm: how can I be sure God will persist for me?
But the worry about scope being inclusive enough “for me” on one hand, or the persistence being guaranteed “for me” on the other (or for other people), leads to a very similar underlying experiential problem:
1.1.) Hard Arm: have I really persisted, or persisted “enough”, and/or in the right way?
1.2.) Soft Arm: was I really serious, did I really do the right things or enough of the right things the right way, to obtain God’s persistence?
The Calv (and Kath) assurance to either of those is that no one has to convince God to save people from sin, He chooses to do so from the beginning. Whoever He has authoritatively chosen, He will certainly persist.
2.1.) Calv: am I really among the scope of those whom God intends to save? How can I be sure?
The Arm (and Kath) assurance to that, is that God originally intends to save everyone from sin. The scope of His salvation certainly includes you and whoever else you may be worried about.
3.) Everyone: are my experiences of assurance only mistakes or delusions?
None of us are really free from this doubt in principle, since our rationales may turn out to be invalid but we haven’t realized it yet, or our data may not be sufficiently accurate, or our experiential experiences (so to speak) may be from something evil disguised in light. We all have to do the best we can under the circumstances.
And maybe our root assurance should be not for our own salvation, or for anyone else’s, but a hope (at least, or a trust if we can manage it) that God is both loving and just, since without one or both those truths we’re all screwed anyway. 
