«По́мнить» is a normal imperfective verb, it means 'to have a memory of' or 'not to forget'. «Ча́сто» 'often' doesn't work with «по́мнить», because you can't keep something in memory 'often', because «по́мнить» means having a memory for the whole period of time. «Ча́сто» means a repeating action, so it doesn't work here: «по́мнить» is not really repeating.
«Вспомина́ть» is an interative of «вспо́мнить», it means 'to be bringing something in memory (over and over), to be recollecing'. It can also mean bringing up some remembrance in conversation or for some other reason. For example, «вспомина́ть до́брым сло́вом» means 'to remember gratefully' (i.e. 'to be bringing in conversation with a kind word'; there's also a quote from Shevchenko «вспомина́ть незлы́м ти́хим сло́вом» 'to bring up in conversation with a kind, quiet word', which is used ironically, meaning someone is rememberd as annoying). «Вспомина́ть» works well with «ча́сто», because it's a repeating action.
Yes, those words are indeed related.
In the past, вос- and вс- were interchangeable (in fact, the archaic forms like воспомина́ть instead of вспомина́ть 'to remember' are still found in the dictionaries). In modern Russian, however, the meaning got separated:
- вспоминать is the verb 'to recall, to remember',
- вспомина́ние is a noun denoting an action: 'recalling, bringing back in memory',
- воспомина́ние is a noun denoting a piece of information remembered: 'a remembrance, a memory'.
Originally they were the same noun (вспоминать or воспоминать were verbs meaning 'to remember', вспоминание and воспоминание denoted either an action or the information remembered), but during the 20th century, воспоминать became obsolete, and the meanings of вспомина́ние and воспомина́ние became separated.
Similar but not quite the same.
"Usually" implies that you do 'x' action the majority of times that you do 'y' action. --> It implies a large proportion; it doesn't (necessarily) indicate frequency.
"Often" (or "frequently") means that you do 'x' action quite a lot. --> It refers only to frequency; it doesn't indicate proportion.
"I usually eat at Luigi's Pizza."
--> You ate at restaurants 6 times this year. You ate at Luigi's Pizza 5 times out of those 6 times.
So when you eat at a restaurant, you usually eat at Luigi's Pizza (majority / large proportion). But you don't eat at restaurants often; only once every 2 months.
"I often eat at Luigi's Pizza."
--> You ate at Luigi's Pizza 12 times this year. However, you actually ate at restaurants 100 times this year.
So even though you often eat at Luigi's Pizza (at least once every month), you ate somewhere else 88 times out of 100. You don't usually at Luigi's Pizza (low proportion).
Hope that makes sense!
«Мы вспо́мним» will mean ‘we will remember’, but it doesn’t combine well with «часто». Because perfective implies an one-time action, and часто means the action has taken place often, so it can’t be one-time.