I was taught that the て form of the verb ある is あって, but I sometimes see the word ありまして acting in the same way. What's the difference and when would you use one or the other?
「ありまして」is just a really polite form of 「あって」.
In the standard polite sentence, only the final verb is put into the polite -ます form, while the rest are in the regular dictionary forms:
While often overkill, it is possible to put the other connecting verbs into the -ます form as well. The resulting「まして」form has the same function as the regular connective -て form, just adding extra politeness- which is why it is common in keigo sentences.
Also, although this isn't really a problem with ありまして, one thing you should be aware of is that you cannot use the -まして in a request with ください.