One major difference between the Tesla Wall Connector and the Tesla Mobile Connector is that the Wall Connector has a NEMA 4 rating, i.e. is fully weather rated for outdoor installation, use. The Mobile Connector is not designed to be continuously exposed to the weather, i.e. rain, sleet, ice or snow. For use inside a garage it would not matter.
If you require a faster charging rate the Wall Connector, when installed on a 60A rated circuit, can charge at up to 240V, 48 amps. That is twice the power of the dryer circuit you are probably using currently. Charging would be approximately twice as fast as with the dryer circuit. The Wall Connector can also be configured for charging using lower amperage circuits.
The NEMA 14-50 receptacle is popular, can accommodate different electrical vehicle service equipment (EVSE) including Tesla and 3rd party EVSEs. Using the 14-50 receptacle and an EVSE configured with a 14-50 plug electric vehicle charging can be performed at up to 32 amps or 40 amps but not 48 amps. One advantage is that if there is ever a problem with the Tesla Mobile Connector the unit can be easily unplugged, replaced without requiring an electrician (or someone with electrical equipment installation skills.) In many states, since 2017, the electrical code now requires a ground fault circuit interrupt (GFCI) device be installed on a circuit for a NEMA 14-50 receptacle that will be used for charging an EV. This requires a GFCI circuit breaker to be installed at the service panel, cost about $100. By comparison, a standard 50A etc. double pole circuit breaker costs just $10. (The Wall Connector, because it is hard wired, with no receptacle, does not require a GFCI circuit breaker.)