Sequential pattern mining: In this, we look into the set of sequences to try to find some interesting subsequences. For example, we can have sequences of customer transactions. We can find a sequential pattern indicating that people who buy a shirt will then buy another shirt. But sequential pattern mining can also be applied to other types of data. Like, one could have a text document, where each sentence can be viewed as a sequence. Then, sequential patterns can be found that is sequences of words that are interesting in text documents.
The motive behind process mining is to analyse the business process. Unlike Sequential pattern mining, Process mining is defined for a specific type of data, which are business processes or other kinds of processes. But since business process logs are sequences of events, one can also apply sequential pattern mining to process logs too. And you could also call that process mining. Thus, there is some overlap between process mining and sequential pattern mining. And although process mining techniques are designed to analyse processes, you may possibly use them on other types of data.
Another difference is that process mining is defined from the perspective of an application, while sequential pattern mining is more general and is defined for analyzing any types of sequences.
In terms of scalability, sequential pattern mining more scalable that Sequential pattern mining.