Alan Turing developed the Turing machine in order to find a mechanical way of solving any mathematical question. This is the
Entscheidungsproblem posed by
Hilbert in 1928.
A Turing machine is capable of performing any mechanical sequence of steps (any algorithm) but I do not believe it is fair to say that they can do everything that a general purpose computer can do.
By definition, you cannot examine a Turing machine's tape until the machine has halted. However, an operating system (OS) on a computer can run indefinitely but that doesn't stop us seeing it's state. In fact, it would be impossible to use an OS without being able to examine it's state while it is running.
Another difference that computers have over Turing machines is user input. Here, I am using user to mean any agent external to the processor. It can be a human user or a piece of hardware. A program sometimes has to halt and wait for input, which may never come. In this situation, a computer can timeout but a Turing machine has no concept of time.
I do not dispute that the core of a computer can be represented by a Turing machine. Any sequence of statements that run uninterrupted can be performed by a Turing machine but I think that a computer is more than simply a machine to step through mechanical procedures. It allows interaction and alteration of those procedures as they run.